Special effect system, image processing method, and symmetrical image generating method

ABSTRACT

Specified image transformation is applied to a first source video signal to generate a main image and symmetric image transformation is applied to a second source video signal same as or different from the first source video signal in accordance with the image transformation applied to the main image and the point, line, or plane specified as a symmetry yardstick to generate a symmetric image point-symmetric, line-symmetric, or plane-symmetric to the main image. Thereby, it is possible to easily obtain a symmetric image symmetric to a main image to which image transformation is applied only by specifying a point, line, or plane serving as a symmetry yardstick.

TECHNICAL FIELD

The present invention relates to a special effect system, particularly to a special effect system capable of easily generating a symmetric image point-symmetric, line-symmetric, or plane-symmetric to a main image to which a special effect is applied.

BACKGROUND ART

Conventionally, in the field of image processing, a special effect such as three-dimensional transformation, rotation, or movement is frequently applied to a source image. Moreover recently, a symmetric image point-symmetric, line-symmetric, or plane-symmetric to a main image generated by applying a special effect to a source image is obtained and it is displayed together with the main image. For example, as shown in FIG. 1, by displaying a main image 1 generated by applying three-dimensional rotational processing to the main image 1 and a symmetric image 1A line-symmetric to the main image 1, it is possible to provide an image 2 in which two persons seemingly talk to each other for audiences.

To generate the above image 2, a conventional special-effect system first obtains the main image 1 by three-dimensionally rotating a first source image by a predetermined angle and displays the image 1 at a predetermined position in the screen shown in FIG. 1. Moreover, the special effect system displays the symmetric image 1A by rotating a second source image by an angle equal to the case of the first source image in the direction opposite to the case of the first source image and thereafter, displaying the second source image at a line-symmetric position designated by an operator. Thereby, the conventional special-effect system obtains the image 2 in which the main image 1 and the symmetric image 1A are displayed at the same time.

In this case, to designate a symmetric position, the operator calculates the symmetric position of the main image 1 and inputs the position to the special effect system through an input unit such as a ten-key pad or slowly inputs positional information through an input unit such as a track ball, joy stick, or mouse to designate a symmetric position by eye measure. Therefore, the conventional special-effect system is problematic in that it takes a lot of time to designate the symmetric position thus burdening the operator. Moreover, to designate the symmetric position by an input unit such as a track ball, joy stick, or mouse, the operator designates the symmetric position by eye measure while viewing the screen in which the symmetric image 1A is displayed. Therefore, there are problems that the symmetric position cannot be accurately designated and therefore, the symmetric image 1A cannot be accurately obtained.

Furthermore, as depicted in FIG. 2, to display the main image 1 and the symmetric image 1A line-symmetric to the main image 1 and obtain an image 3 in which the symmetric image 1A is moved in accordance with the movement of the main image 1 (note the symmetrical movement of image 1A as the main image 1 moves in FIG. 2), there are problems that the symmetric position of the main image 1 must be designated each time and thereby, a lot of time is required.

DISCLOSURE OF THE INVENTION

The present invention is made to solve the above problems and i its object is to provide a special effect system capable of easily obtaining a symmetric image by designating a point, line, or plane serving as a symmetry yardstick without designating a symmetric position by an operator.

To solve the above problems, the present invention uses a special effect system for applying image transformation to an input source video signal, which comprises first image processing means for applying specified image transformation to an input first source video signal to generate the video signal of a main image and second image processing means for generating the video signal of a symmetric image point-symmetric, line-symmetric, or plane-symmetric to the main image by receiving a second source video signal same as or different from the first source video signal and applying symmetric image transformation to the second source video signal in accordance with the image transformation applied to the main image and a point, line, or plane specified as a symmetry yardstick. Thereby, only by specifying a point, line, or plane serving as a symmetry yardstick, it is possible to easily obtain a symmetric image symmetric to a main image to which image transformation is applied.

Moreover, the present invention uses a special effect system for applying image transformation to an input source video signal, which comprises control means for computing a first matrix parameter for image transformation in accordance with a specified special-effect parameter and computing a second matrix parameter for obtaining a point-symmetric line-symmetric or plane-symmetric image in accordance with the special effect parameter and a point, line, or plane serving as a specified symmetry yardstick, a first read address generation means for generating a first read address for image transformation in accordance with the first matrix parameter received, second read address generation means for generating a second read address for obtaining a point-symmetric, line-symmetric, or plane-symmetric image in accordance with the second matrix parameter received, first memory means for generating the video signal of a main image by successively storing the input first source video signal in an internal storage area and reading the first source video signal from the storage area specified by the first read address in order and thereby, applying image transformation specified by the special effect parameter to the first source video signal, and second memory means for generating the video signal of a symmetric image point-symmetric, line-symmetric, or plane-symmetric to the main image by receiving a second source video signal same as or different from the first source video signal and successively storing the second source video signal in an internal storage area and reading the second source video signal from the storage are a specified by the second read address.

Thus, it is possible to easily obtain a symmetric image symmetric to a main image to which image transformation is applied only by specifying a point, line, or plane serving as a symmetry yardstick by computing a second matrix parameter in accordance with a special effect parameter to be applied to a main image and a point, line, or plane specified as a symmetry yardstick, making the matrix parameter work on a second read address, and reading a second source video signal from a memory in accordance with the second read address.

BRIEF DESCRIPTION OF DRAWINGS

FIG. 1 is a schematic screen diagram for explaining the background art;

FIG. 2 is a schematic screen diagram for explaining the background art;

FIG. 3 is a block diagram showing a special effect system of the present invention;

FIG. 4 is a schematic screen diagram for explaining a symmetric image obtained by a special effect system of the present invention;

FIG. 5 is a schematic view for explaining a world coordinate system defined for a special effect system of the present invention;

FIGS. 6A and 6B are schematic diagrams for explaining three-dimensional image transformation processing;

FIGS. 7A and 7B are schematic diagrams showing the relation between address on a frame memory and address on a monitor screen;

FIG. 8 is a schematic diagram for explaining two-dimensional point-symmetric transformation;

FIG. 9 is a schematic diagram for explaining two-dimensional point-symmetric transformation;

FIG. 10 is a schematic diagram for explaining two-dimensional point-symmetric transformation.

FIG. 11 is a schematic diagram for explaining two-dimensional point-symmetric transformation;

FIG. 12 is a schematic diagram for explaining three-dimensional point-symmetric transformation;

FIG. 13 is a schematic diagram for explaining three-dimensional point-symmetric transformation;

FIG. 14 is a schematic diagram for explaining three-dimensional point-symmetric transformation;

FIG. 15 is a schematic diagram for explaining three-dimensional point-symmetric transformation;

FIG. 16 is a schematic diagram for explaining two-dimensional line-symmetric transformation;

FIG. 17 is a schematic diagram for explaining two-dimensional line-symmetric transformation;

FIG. 18 is a schematic diagram for explaining two-dimensional line-symmetric transformation;

FIG. 19 is a schematic diagram for explaining two-dimensional line-symmetric transformation;

FIG. 20 is a schematic diagram for explaining two-dimensional line-symmetric transformation;

FIG. 21 is a schematic diagram for explaining two-dimensional line-symmetric transformation;

FIG. 22 is a schematic diagram for explaining three-dimensional line-symmetric transformation;

FIG. 23 is a schematic diagram for explaining three-dimensional line-symmetric transformation;

FIG. 24 is a schematic diagram for explaining three-dimensional line-symmetric transformation;

FIG. 25 is a schematic diagram for explaining three-dimensional line-symmetric transformation;

FIG. 26 is a schematic diagram for explaining plane-symmetric transformation;

FIG. 27 is a schematic diagram for explaining plane-symmetric transformation;

FIG. 28 is a schematic diagram for explaining plane-symmetric transformation;

FIG. 29 is a schematic diagram for explaining plane-symmetric transformation;

FIG. 30 is a schematic diagram for explaining plane-symmetric transformation;

FIG. 31 is a schematic diagram for explaining plane-symmetric transformation;

FIG. 32 is a schematic diagram for explaining plane-symmetric transformation;

FIG. 33 is a schematic diagram for explaining plane-symmetric transformation;

FIG. 34 is a schematic diagram for explaining plane-symmetric transformation;

FIG. 35 is a schematic diagram for explaining plane-symmetric transformation based on a plane parallel with z-axis;

FIG. 36 is a schematic diagram for explaining an effect file;

FIG. 37 is a schematic diagram for explaining a case in which a plurality of key frames are entered in an effect file;

FIG. 38 is a screen diagram for explaining a case of continuously moving a main image in accordance with key frames;

FIG. 39 is a schematic diagram for explaining an effect file related to a symmetric image; and

FIG. 40 is a screen diagram for explaining a case of moving a main image and a symmetric image symmetrically to y-axis.

BEST MODE FOR CARRYING OUT THE INVENTION

(1) Entire structure of special effect system

First, a special effect system of the present invention is described below by referring to FIG. 3.

In FIG. 3, symbol 10 denotes a special effect system of the present invention as a whole, which has a first image processing section 11 for generating a main image 5 to which a special effect such as three-dimensional transformation, rotation, or movement is applied and a second image processing section 12 for generating a symmetric image 5A point-symmetric, line-symmetric, or plane-symmetric to the main image 5. The special effect system 10 is provided with a CPU (Central Processing Unit) 13 so as to generate a video signal V_(out) comprising the main image 5 and symmetric image 5A by controlling first and second image processing sections 11 and 12 and other circuit blocks by the CPU 13.

The CPU 13 first receives the input information input by operating the operation unit of a control panel 14 by an operator through an interface circuit (I/F) 14A and a data bus 15 to control each circuit block in accordance with the input information. In this case, parameters related to a special effect to be applied to the main image 5 and data for specifying the symmetry yardstick of the symmetric image 5A are input as the input information. Moreover, the CPU 13 enters these input data values in an effect file (to be described later in detail) and stores the effect file in a RAM (Random Access Memory) 17. In this connection, the CPU 13 displays the input information on an indicator 14B and thereby, the operator can confirm the input contents by viewing the screen of the indicator 14B.

The CPU 13 computes matrix parameters b₁₁ to b₃₃ of a transformation matrix required to generate the main image 5 in the first image processing section 11 in accordance with the parameters related to a special effect in the effect file and supplies the parameters to the first image processing section 11. Moreover, the CPU 13 computes matrix parameters b₁₁′ to b₃₃′ of a transformation matrix required to generate the symmetric image 5A in the second image processing section 12 in accordance with the data for specifying the parameters and symmetry yardstick related to a special effect in the effect file and supplies the parameters to the second image processing section 12. Moreover, the CPU 13 uses the RAM 17 as a working memory to store the various data values generated to compute the matrix parameters b₁₁ to b₃₃ and b₁₁′ to b₃₃′ and the computed matrix parameters b₁₁ to b₃₃ and b₁₁′ to b₃₃′ in the RAM 17.

Furthermore, the CPU 13 controls a screen address generation circuit 18 in accordance with the data related to the display position of the main image 5 among the parameters related to a special effect in the effect file and thereby, makes the circuit 18 generate a screen address (X_(S), Y_(S)) corresponding to the display position of the main image 5. In this connection, the screen address generation circuit 18 includes a signal generation circuit for generating horizontal and vertical synchronizing signals, generates the screen address (X_(S), Y_(S)) corresponding to the display position of the main image 5 by addressing the entire screen of a monitor screen 16 in order of luster scan in accordance with the horizontal and vertical synchronizing signals generated by the signal generation circuit and outputs the screen address to the first and second image processing sections 11 and 12.

Furthermore, the CPU 13 is operated in accordance with an operation program stored in ROM (Read Only Memory) 19 and thereby, controls the above-described circuit blocks and computes the matrix parameters b₁₁ to b₃₃ and b₁₁′ to b₃₃′.

A first source video signal V₁ and a first key signal K₁ for keying the first source video signal V₁ are input to the first image processing section 11. The first image processing section 11 is provided with two frame memories 11A and 11B so as to input the input first source video signal V₁ to the frame memory 11A and the first key signal K₁ to the frame memory 11B. The frame memory 11A receives a sequential write address from a not-illustrated write address generation circuit and sequentially stores the input first source video signal V₁ in its internal storage area in accordance with the write address. Similarly, the frame memory 11B receives a sequential write address from a not-illustrated write address generation circuit and sequentially stores the input first key signal K₁ in its internal storage area in accordance with the write address.

In this connection, at this stage, the first source video signal V₁ and the first key signal K₁ are only stored but image transformation processing is not applied to the first source video signal V₁ or the first key signal K₁.

A read address generation circuit 11C generates a read address (X_(M), Y_(M)) for generating the main image 5 by performing the image transformation processing designated by an operator in accordance with the screen address (X_(S), Y_(S)) supplied from the screen address generation circuit 18 and the matrix parameters b₁₁ to b₃₃ of the transformation matrix supplied from the CPU 13, supplies the address data D_(AD1) of the integer part of the read address (X_(M), Y_(M)) to the frame memories 11A and 11B, and supplies the address data D_(AD2) of the fraction part of the read address to interpolation circuits 11D and 11E.

The frame memory 11A reads a video signal from the storage area specified by the address data D_(AD1) and outputs the read video signal V₃ to the interpolation circuit 11D. The interpolation circuit 11D calculates an interpolation coefficient in accordance with the address data D_(AD2) and applies pixel interpolation processing to the read video signal V₃ in accordance with the interpolation coefficient.

The following is the reason for performing the interpolation processing by the interpolation circuit 11D. The read address (X_(M), Y_(M)) generated by the read address generation circuit 11C is not always an integer but it may include a decimal. If the read address (X_(M), Y_(M)) includes a decimal, it is impossible to perform the read operation because no decimal address is present in the frame memory 11A. Therefore, the read address (X_(M), Y_(M)) is divided into an integer part and a decimal part to obtain a video signal corresponding to the read address (X_(M), Y_(M)) including a decimal by applying interpolation processing to the video signal V₃ read in accordance with the integer part. Thereby, even when the read address (X_(M), Y_(M)) includes a decimal, it is possible to obtain a video signal corresponding to the address (X_(M), Y_(M)).

Thus, a video signal V₄ to which the image transformation processing designated by an operator is applied is generated (that is, the video signal V₄ of the main image 5 is generated) by reading the video signal V₃ from the frame memory 11A in accordance with the integer part of the read address (X_(M), Y_(M)) generated by the read address generation circuit 11C and applying the interpolation processing to the video signal V₃ read in accordance with the decimal part of the read address (X_(M), Y_(M)).

Similarly, the frame memory 11B reads a key signal from the storage area specified by the address data D_(AD1) and outputs the read key signal K₃ to the interpolation circuit 11E. The interpolation circuit 11E calculates an interpolation coefficient in accordance with the address data D_(AD2) and applies the pixel interpolation processing to the read key signal K₃ in accordance with the interpolation coefficient. In this connection, the reason of performing the interpolation processing by the interpolation circuit 11E is the same as the above reason.

Thus, a key signal K₄ to which the image transformation processing same as the case of the video signal V₄ is applied is generated by reading the key signal K₃ from the frame memory 11B in accordance with the integer part of the read address (X_(M), Y_(M)) generated by the read address generation circuit 11C and applying the interpolation processing to the read key signal K₃ in accordance with the decimal part of the read address (X_(M), Y_(M)).

A second source video signal V₂ same as or different from the first source video signal V₁ and a second key signal K₂ for keying the second source video signal V₂ are input to the second image processing section 12. The second image processing section 12 is provided with two frame memories 12A and 12B so as to input the input second source video signal V₂ to the frame memory 12A and the second key signal K₂ to the frame memory 12B. The frame memory 12A sequentially stores the input second source video signal V₂ in its internal storage area in accordance with a sequential write address supplied from a not-illustrated write address generation circuit. Similarly, the frame memory 12B receives a sequential write address from a not-illustrated write address generation circuit and sequentially stores the input second key signal K₂ in its internal storage area in accordance with the write address.

In this connection, the second image processing section 12 only stores the second source video signal V₂ and second key signal K₂ at this stage but the image transformation processing is not applied yet to the second source video signal V₂ or second key signal K₂.

The read address generation circuit 12C generates a read address (X_(M)′, Y_(M)′) for generating the symmetric image 5A for the main image 5 generated by the first image processing section 11 in accordance with the screen address (X_(S), Y_(S)) supplied from the screen address generation circuit 18 and the matrix parameters b₁₁′ to b₃₃′ of the transformation matrix supplied from the CPU 13, supplies the address data D_(AD1)′ of the integer part of the read address (X_(M)′, Y_(M)′) to the frame memories 12A and 12B, and supplies the address data D_(AD2)′ of the decimal part of the address (X_(M)′, Y_(M)′) to the interpolation circuits 12D and 12E.

The frame memory 12A reads a video signal from the storage area specified by the address data D_(AD1)′ and outputs the read video signal V₅ to the interpolation circuit 12D. The interpolation circuit 12D calculates an interpolation coefficient in accordance with the address data D_(AD2)′ and applies the pixel interpolation processing to the read video signal V₅ in accordance with the interpolation coefficient. In this connection, the reason of performing interpolation processing by the interpolation circuit 12D is the same as the case of the first image processing section 11.

Thus, a video signal V₆ of the symmetric image 5A for the main image 5 to which image transformation is applied is generated by reading the video signal V₅ from the frame memory 12A in accordance with the integer part of the read address (X_(M)′, Y_(M)′) generated by the read address generation circuit 12C and applying the interpolation processing to the read video signal V₅ from the frame memory 12A in accordance with the decimal part of the read address (X_(M)′, Y_(M)′).

Similarly, the frame memory 12B reads a key signal from the recording area specified by the address data D_(AD1)′ and outputs the read key signal K₅ to the interpolation circuit 12E. The interpolation circuit 12E calculates an interpolation coefficient in accordance with the address data D_(AD2)′ and applies the pixel interpolation processing to the read key signal K₅ in accordance with the interpolation coefficient. In this connection, the reason, of performing the interpolation processing by the interpolation circuit 12E is the same as the case of the first image processing section 11.

Thus, a key signal K₆ to which the image transformation same as the case of the video signal V₆ is applied is generated by reading the key signal K₅ from the frame memory 12B in accordance with the integer part of the read address (X_(M)′, Y_(M)′) generated by the read address generation circuit 12C and applying the interpolation processing to the read key signal K₅ in accordance with the decimal part of the read address (X_(M)′, Y_(M)′).

Thus, the video signals V₄ and V₆ and key signals K₄ and K₆ image-transformed by the first and second image processing sections 11 and 12 are output to a first mixer 20.

The first mixer 20 is a circuit for mixing the video signal V₄ of the main image 5 with the video signal V₆ of the symmetric image 5A and moreover mixing the key signal K₄ of the main image 5 with the key signal K₆ of the symmetric image 5A. Specifically, the first mixer 20 mixes the video signal V₄ with the video signal V₆ and mixes the key signal K₄ with the key signal K₆ by performing the processing shown by the following expression (1) and outputs a resultingly-obtained mixed video signal V_(MIX) and mixed key signal K_(MIX) to a second mixer 21.

V _(MIX) =K ₄ V ₄ +K ₆ V ₆

K _(MIX) =K ₄ +K ₆   (1)

The second mixer 21 is a circuit for mixing the mixed video signal V_(MIX) with a background video signal V_(BK) supplied from an external unit in accordance with the mixed key signal K_(MIX). Specifically, the second mixer 21 mixes the mixed video signal V_(MIX) with the background video signal V_(BK) by performing the processing shown by the following expression (2).

V _(OUT) =K _(MIX) V _(MIX)+(1−K _(MIX))V _(BK)   (2)

Thereby, a video signal V_(OUT) comprising the main image 5 and the symmetric image 5A is generated. The video signal V_(OUT) thus generated is output to an external unit and also displayed on the monitor screen 16.

Thus, in the case of the special effect system 10, an image 6 in which two persons seemingly talk each other shown in FIG. 4 is obtained by inputting the video signal of a predetermined person as the first source video signal V₁, the video signal of a person different from the former person as the second source video signal V₂, and a parameter of a special effect for rotating the first source video signal V₁ by a predetermined angle, and data for specifying line symmetry through the control panel 14. In this case, an operator can easily obtain the symmetric image 5A only by inputting the special effect parameter and the data showing a symmetry yardstick through the control panel 14. It is possible to greatly reduce the labor of the operator compared to the conventional case of calculating and inputting a symmetric position or gradually inputting positional information through an input unit such as a track ball, joy stick, or mouse to designate a symmetric position by eye measure. Moreover, it is possible to obtain the accurate symmetric image 5A because a parameter can be accurately input compared to the conventional case of inputting positional information through an input unit such as a track ball, joy stick, or mouse by eye measure.

(2) Method for generating basic algorithm for three-dimensional image transformation and read address used for the image transformation

In this section, a method for generating a basic algorithm for three-dimensional image transformation such as transformation, rotation, or movement and a read address used to perform the image transformation is described.

(2-1) Definition of world coordinate system

First, a world coordinate system used for explanation of the present invention while referring to FIG. 5. The world coordinate system represents a three-dimensional rectangular coordinate system comprising x-, y-, and z-axes. That is, as shown in FIG. 5, x-axis is defined as the horizontal (right-and-left) direction of the screen plane and y-axis is defined as the vertical (up-and-down) direction of the screen plane by assuming that the screen plane of the monitor screen 16 is present on the xy plane defined by x-axis and y-axis perpendicular to x-axis.

Moreover, the depth direction of the screen plane is defined as the positive direction of z-axis perpendicular to the xy plane and the this side of the screen plane, that is, the side where a viewpoint PZ for viewing the screen plane is present is defined as the negative direction of z-axis.

Moreover, it is defined that the center of the screen plane coincides with the origin of the world coordinate system comprising these x-axis, y-axis, and z-axis.

Continuous virtual coordinate values are set to x-axis from the inside (origin) of the screen area toward the right and left outward directions and virtual coordinate values between “−4” and “+4” are set to x-axis in the screen area from left to right sides when viewing the screen plane from the viewpoint PZ.

Moreover, continuous coordinate values are set to y-axis from the inside (origin) of the screen area toward upper and lower outward directions and virtual coordinate values between “−3” and “+3” are set to y-axis in the screen area from lower to upper sides when viewing the screen plane from the viewpoint PZ.

Furthermore, the viewpoint position PZ of the operator is virtually set to a position where the coordinate value of the position PZ is “−16” on z-axis.

(2—2) Basic algorithm of three-dimensional image transformation

In this section, a basic algorithm for generating the video signal V₃ to which three-dimensional image transformation is applied from the source video signal V₁ is described by referring to FIGS. 6A and 6B.

The source video signal V₁ is directly stored in the frame memory 11A without being image-transformed. Because the source video signal V₁ is a two-dimensional video signal, it is present on the screen plane of the monitor screen 16 in a three-dimensional space as shown in FIGS. 6A and 6B.

When the operator designates the three-dimensional image transformation comprising rotation of approx. 45° about y-axis and, translation in the positive direction of z-axis to the source video signal V₁ as shown in FIGS. 6A and 6B, it is possible to generate a video signal V₁′ to which the designated image transformation is applied by multiplying each pixel of the source video signal V₁ by a three-dimensional transformation matrix T₀ corresponding to the image transformation.

The three-dimensional transformation matrix T₀ can be generally shown by the following expression (3). $\begin{matrix} {T_{0} = \begin{bmatrix} r_{11} & r_{12} & r_{13} & 0 \\ r_{21} & r_{22} & r_{23} & 0 \\ r_{31} & r_{32} & r_{33} & 0 \\ l_{x} & l_{y} & l_{z} & s \end{bmatrix}} & (3) \end{matrix}$

Transformation parameters r₁₁ to r₃₃ used for the transformation matrix T₀ are parameters including an element for rotating the source video signal V₁ about x-, y-, or z-axis, an element for enlarging or reducing the scale of the source video signal V₁ in the x-, y-, and z-axis direction, and an element for skewing the source video signal V₁ in the x-, y-, or z-axis direction. Moreover, transformation parameters l_(x), l_(y), and l_(z) are parameters including an element for translating the source video signal V₁ in the x-, y-, or z-axis direction and a parameter s is a parameter including an element for uniformly enlarging or reducing the whole of the source video signal V₁ in each axis direction of three dimensions.

The transformation matrix T₀ is a matrix of four rows and four columns because a coordinate system for rotational transformation and the like and a coordinate system for translational transformation and enlargement/reduction transformation are shown in one coordinate system. In general, this coordinate system is referred to as homogeneous coordinate system.

Because the video signal V₁′ coordinate-transformed into a three-dimensional space by making the three-dimensional transformation matrix T₀ work is finally displayed on the screen plane of the monitor screen 16, it is necessary to project the signal V₁′ on a screen plane based on the viewpoint of an operator. That is, as shown in FIG. 6B, it is necessary to obtain the video signal V₃ through-viewed on the xy plane when viewing the video signal V₁′ in a three-dimensional space from the virtual viewpoint PZ on z-axis. This projection processing can be executed by making a perspective transformation matrix P₀ work. That is, by multiplying each pixel of the video signal V₁′ by the perspective transformation matrix P₀, it is possible to obtain the video signal V₃ obtained by through-viewing the video signal V₁′ present in a three-dimensional space onto an xy plane.

The perspective transformation matrix P₀ is generally shown by the following expression (4). $\begin{matrix} {P_{0} = \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & P_{z} \\ 0 & 0 & 0 & 1 \end{bmatrix}} & (4) \end{matrix}$

The parameter P_(Z) of the perspective transformation matrix P₀ is a perspective value for applying the perspective when projecting the video signal V₁′ onto an xy plane. In general, a value “116” is set as the reference value of the perspective value P_(Z). This represents that the z-coordinate value of the virtual viewpoint PZ is “−16”. Moreover, an operator can set the perspective value P_(Z) to a desired value.

Thus, by coordinate-transforming the source video signal V₁ into a three-dimensional space by making the transformation matrix T₀ corresponding to the image transformation designated by an operator work on the signal V₁ and performing the projection processing from the three-dimensional space to an xy plane, it is possible to apply the desired image transformation designated by the operator to the source video signal V₁ and generate the video signal V₃ to which the image transformation is applied.

The contents of the image transformation processing described above are summarized as shown below. That is, the image transformation processing comprises the spatial image transformation step of obtaining the three-dimensional-transformation video signal V₁′ from the source video signal V₁ in accordance with the three-dimensional transformation matrix T₀ and the perspective transformation step of obtaining the perspective-transformation video signal V₃ from the three-dimensional-transformation video signal V₁′ in accordance with the perspective transformation matrix P₀. Therefore, a transformation matrix T for obtaining the video signal V₃ from the source video signal V₁ at a stroke is shown by the following expression (5) in accordance with a multiplication expression between the three-dimensional-transformation matrix T₀ and the perspective transformation matrix P₀. $\begin{matrix} \begin{matrix} {T = {T_{0} \cdot P_{0}}} \\ {= {\begin{bmatrix} r_{11} & r_{12} & r_{13} & 0 \\ r_{21} & r_{22} & r_{23} & 0 \\ r_{31} & r_{32} & r_{33} & 0 \\ l_{x} & l_{y} & l_{z} & s \end{bmatrix}\begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & P_{z} \\ 0 & 0 & 0 & 1 \end{bmatrix}}} \\ {= \begin{bmatrix} r_{11} & r_{12} & r_{13} & {r_{13}P_{z}} \\ r_{21} & r_{22} & r_{23} & {r_{23}P_{z}} \\ r_{31} & r_{32} & r_{33} & {r_{33}P_{z}} \\ l_{x} & l_{y} & l_{z} & {{l_{z}P_{z}} + s} \end{bmatrix}} \end{matrix} & (5) \end{matrix}$

Therefore, by multiplying each pixel of the source video signal V₁ by the transformation matrix T shown by the expression (5), it is possible to generate the video signal V₃ to which desired image transformation is applied.

When generating the video signal V₃, the special effect system 10 of the present invention does not multiply the source video signal V₁ by the transformation matrix T but actually, the system 10 generates the video signal V₃ by obtaining a read address to which the image transformation shown by the transformation matrix T will be applied and reading the source video signal V₁ from the frame memory 11A in accordance with the read address.

That is, the special effect system 10 of the present invention generates the video signal V₃ to which image transformation is applied by successively writing the source video signal V₁ in the frame memory 11A and reading the source video signal V₁ in accordance with a read address to which the image transformation shown by the transformation matrix T will be applied.

The source video signal V₁ to be written in the frame memory 11A of the special effect system 10 and the video signal V₃ read from the frame memory 11A are two-dimensional video data and moreover, the frame memory 11A is a memory for storing two-dimensional data. Therefore, in the case of the operation of an read address used for the read operation from the frame memory 11A, a parameter for computing z-axis-directional data in a three-dimensional space is not practically used. Therefore, the parameter at the third column and third row for computing z-axis-directional data in the transformation matrix T shown in the expression (5) is unnecessary.

That is, when it is assumed that a transformation matrix to be actually required for the operation of a read address is T₃₃, the transformation matrix T₃₃ becomes a matrix obtained by excluding the third column and third row in the expression (5) and it is shown by the following expression (6). $\begin{matrix} {T_{33} = \begin{bmatrix} r_{11} & r_{12} & {r_{13}P_{z}} \\ r_{21} & r_{22} & {r_{23}P_{z}} \\ l_{x} & l_{y} & {{l_{z}P_{z}} + s} \end{bmatrix}} & (6) \end{matrix}$

(2-3) Method for generating read address

Then, a method for generating a read address used for the read operation from the frame memory 11A is described below. First, the relation between position vector on the frame memory 11A and position vector on the monitor screen 16 is described below by referring to FIGS. 7A and 7B.

It is assumed that a two-dimensional address on the frame memory 11A is (X_(M), Y_(M)), position vector on the frame memory 11A is [X_(M) Y_(M)], and an address on the monitor screen 16 is (X_(S), Y_(S)) and position vector on the screen 16 is [X_(S) Y_(S)]. Then, two-dimensional position vector [X_(M) Y_(M)] on the frame memory 11A can be shown as vector [x_(m) y_(m) H₀] in terms of a homogeneous coordinate system and the position vector [X_(S) Y_(S)] on the monitor screen 16 can be shown by vector [x_(s) y_(s) 1] in terms of the homogeneous coordinate system. The parameter “H₀” of the homogenous coordinate system is a parameter for showing the enlargement/reduction ratio of the magnification of vector.

By making the transformation matrix T₃₃ work on the position vector [x_(m) y_(m) H₀] on the frame memory 11A, the position vector [x_(m) y_(m) H₀] on the frame memory 11A is transformed into position vector [x_(s) y_(s) 1] on the monitor screen 16. Therefore, the relation between the position vector [x_(m) y_(m) H₀] on the frame memory 11A and the position vector [x_(s) y_(s) 1] on the monitor screen 16 is shown by the following expression (7).

[x _(s) y _(s) 1]=[x _(m) x _(m) H ₀ ]·T ₃₃   (7)

The relation between the parameter “H₀” of the homogeneous coordinate system used for the position vector [x_(m) y_(m) H₀] on the frame memory 11A and the parameter “1” of the homogeneous coordinate system used for the position vector [x_(s) y_(s) 1] on the monitor screen 16 represents that the position vector [x_(m) y_(m)] on the frame memory 11A is transformed into the position vector [x_(s) y_(s)] on the monitor screen 16 by the transformation matrix T₃₃ and the enlargement/reduction ratio “H₀” of the position vector [x_(m) y_(m)] on the frame memory 11A is transformed so that the ratio “H₀” becomes equal to the enlargement/reduction ratio “1” of the position vector [x_(s) y_(s)] on the monitor screen 16.

Therefore, the expression (7) is a relational expression for obtaining a point on the monitor screen 16 corresponding to a point on the frame memory 11A by the transformation matrix T₃₃.

The special effect system 10 of the present invention applies spatial image transformation to the source video signal V₁ by storing the source video signal V₁ in the frame memory 11A under the state before transformation, reading a point on the frame memory 11A corresponding to a point on the monitor screen 16 obtained by the transformation matrix T₃₃, and specifying it by a read address. That is, image transformation is not performed when writing the signal V₁ in the frame memory 11A but it is performed when the signal V₁ is read from the frame memory 11A.

In the case of the special effect system 10, it is necessary to obtain a point on the frame memory 11A corresponding to a point on the monitor screen 16 instead of performing the operation according to the expression (7) for obtaining a point on the monitor screen 16 corresponding to a point on the frame memory 11A. Therefore, it is necessary to obtain a point on the frame memory 11A corresponding to a point on the monitor screen 16 by transforming the expression (7) and using a relational expression shown by the following expression (8). In this connection, a transformation matrix T₃₃ ⁻¹ is an inverse matrix of the transformation matrix T₃₃.

[x _(m) y _(m) H ₀ ]=[x _(s) y _(s) 1]·T ₃₃ ⁻¹  (8)

Then, a method for actually obtaining the two-dimensional position vector [X_(M) Y_(M)] on the frame memory 11A is described below in accordance with the above idea. First, the transformation matrix T₃₃ is shown by parameters a₁₁ to a₃₃ as shown by the following expression (9). $\begin{matrix} {T_{33} = {\begin{bmatrix} r_{11} & r_{12} & {r_{13}P_{z}} \\ r_{21} & r_{22} & {r_{23}P_{z}} \\ l_{x} & l_{y} & {{l_{z}P_{z}} + s} \end{bmatrix} = \begin{bmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{bmatrix}}} & (9) \end{matrix}$

Then, the inverse matrix T₃₃ ⁻¹ is shown by parameters b₁₁ to b₃₃ as shown by the following expression (10). $\begin{matrix} {{T_{33}^{- 1} = {\begin{bmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{bmatrix}^{- 1} = \begin{bmatrix} b_{11} & b_{12} & b_{13} \\ b_{21} & b_{22} & b_{23} \\ b_{31} & b_{32} & b_{33} \end{bmatrix}}}{Where}{b_{ij} = \frac{a_{ji}}{\det \left( T_{33} \right)}}} & (10) \end{matrix}$

The inverse matrix T₃₃ ⁻¹ thus defined is substituted for the above expression (8) and developed to obtain the following expression (11) $\begin{matrix} \begin{matrix} {\begin{matrix} \left\lbrack x_{m} \right. & y_{m} & \left. H_{0} \right\rbrack \end{matrix} = \quad {\begin{matrix} \left\lbrack x_{s} \right. & y_{s} & \left. 1 \right\rbrack \end{matrix}\begin{bmatrix} b_{11} & b_{12} & b_{13} \\ b_{21} & b_{22} & b_{23} \\ b_{31} & b_{32} & b_{33} \end{bmatrix}}} \\ {= \quad \left\lbrack \begin{matrix} {{b_{11}x_{s}} + {b_{21}y_{s}} + b_{31}} & {{b_{12}x_{s}} + {b_{22}y_{s}} + b_{32}} \end{matrix} \right.} \\ \left. \quad {{b_{13}x_{s}} + {b_{23}y_{s}} + b_{33}} \right\rbrack \end{matrix} & (11) \end{matrix}$

Each element of the position vector [x_(m) y_(m) H₀] on the frame memory 11A is shown by the following expression (12) in accordance with the expression (11).

x _(m) =b ₁₁ x ₃ +b ₂₁ y _(s) +b ₃₁

y _(m) =b ₁₂ x _(s) +b ₂₂ y _(s) +b ₃₂

H ₀ =b ₁₃ x _(s) +b ₂₃ y ₃ +b ₃₃   (12)

In this case, to transform the position vector [x_(m) y_(m) H₀] according to the homogeneous coordinate system on the frame memory 11A into the two-dimensional position vector [X_(M) Y_(M)] on the frame memory 11A, it is necessary to conform to the following procedure. That is, the parameter “H₀” used to transform the two-dimensional position vector [X_(M) Y_(M)] into a homogeneous coordinate system is a parameter showing the enlargement/reduction ratio of the position vector (x_(m) y_(m)) of the homogeneous coordinate system. Therefore, to transform the position vector of a homogeneous coordinate system into a two-dimensional position vector, it is necessary to normalize the parameters “x_(m)” and “y_(m)” showing the direction of the position vector of the homogeneous coordinate system enlargement/reduction ratio of the position vector of the homogeneous coordinate system with the parameter “H₀” showing the enlargement/reduction ratio of the position vector of the homogeneous coordinate system. Therefore, it is possible to obtain the parameters “X_(M)” and “Y_(M)” of a two-dimensional position vector on the frame memory 11A from the following expression (13).

X_(M)=x_(m)/H₀

Y_(M)=y_(m)/H₀  (13)

Moreover, it is possible to transform the position vector [x_(s) y_(s) 1] according to the homogeneous coordinate system on the monitor screen 16 into the two-dimensional position vector [X_(S) Y_(S)] with the same idea. Therefore, it is necessary to normalize the parameters “x_(s)” and “y_(s)” showing the direction of the position vector of the homogeneous coordinate system with the parameter “1” showing the enlargement/reduction ratio of the position vector of the homogeneous coordinate system. Therefore, it is possible to obtain the parameters “X_(S)” and “Y_(S)” of a two-dimensional position vector on, the monitor screen 16 by the following expression (14).

X_(S)=x_(s)

Y_(S)=y_(s)  (14)

Thus, by substituting the expressions (11) and (14) for the expression (13), the parameters “X_(M)” and “Y_(M)” of the two-dimensional position vector on the frame memory 11A can be shown by the following expressions (15) and (16). $\begin{matrix} \begin{matrix} {X_{M} = \frac{x_{m}}{H_{0}}} \\ {= \frac{{b_{11}x_{s}} + {b_{21}y_{s}} + b_{31}}{{b_{13}x_{s}} + {b_{23}y_{s}} + b_{33}}} \\ {= \frac{{b_{11}X_{S}} + {b_{21}Y_{S}} + b_{31}}{{b_{13}X_{S}} + {b_{23}Y_{S}} + b_{33}}} \end{matrix} & (15) \\ \begin{matrix} {Y_{M} = \frac{y_{m}}{H_{0}}} \\ {= \frac{{b_{12}x_{s}} + {b_{22}y_{s}} + b_{32}}{{b_{13}x_{s}} + {b_{23}y_{s}} + b_{33}}} \\ {= \frac{{b_{12}X_{S}} + {b_{22}Y_{S}} + b_{32}}{{b_{13}X_{S}} + {b_{23}Y_{S}} + b_{33}}} \end{matrix} & (16) \end{matrix}$

It is possible to obtain the position vector [X_(M) Y_(M)] on the frame memory 11A by the expressions (15) and (16) and moreover, obtain the read address (X_(M), Y_(M)) on the frame memory 11A.

Then, the parameters b₁₁ to b₃₃ of the inverse matrix T₃₃ ⁻¹ are obtained. It is possible to the parameters b₁₁ to b₃₃ of the inverse matrix T₃₃ ⁻¹ can be shown by the following expressions (17) to (25). $\begin{matrix} {b_{11} = \frac{{{- a_{32}}a_{23}} + {a_{22}a_{33}}}{W_{1}}} & (17) \\ {b_{12} = \frac{{a_{32}a_{13}} - {a_{12}a_{33}}}{W_{1}}} & (18) \\ {b_{13} = \frac{{{- a_{22}}a_{13}} + {a_{12}a_{23}}}{W_{1}}} & (19) \\ {b_{21} = \frac{{a_{31}a_{23}} - {a_{21}a_{33}}}{W_{1}}} & (20) \\ {b_{22} = \frac{{{- a_{31}}a_{13}} + {a_{11}a_{33}}}{W_{1}}} & (21) \\ {b_{23} = \frac{{a_{21}a_{13}} - {a_{11}a_{23}}}{W_{1}}} & (22) \\ {b_{31} = \frac{{{- a_{22}}a_{31}} + {a_{21}a_{32}}}{W_{1}}} & (23) \\ {b_{32} = \frac{{a_{12}a_{31}} - {a_{11}a_{32}}}{W_{1}}} & (24) \\ {b_{33} = \frac{{{- a_{12}}a_{21}} + {a_{11}a_{22}}}{W_{1}}} & (25) \end{matrix}$

In the above expression, a parameter W₁ is a value shown by the following expression (26).

W ₁ −a ₂₂ a ₃₁ a ₁₃ +a ₂₁ a ₃₂ a ₁₃ +a ₁₂ a ₃₁ a ₂₃ −a ₁₁ a ₃₂ a ₂₃ −a ₁₂ a ₂₁ a ₃₃ +a ₁₁ a ₂₂ a ₃₃   (26)

In this case, values of the parameters a₁₁ to a₃₃ are shown by the following expressions (27) to (29) in accordance with the relational expression (9).

a₁₁=r₁₁, a₁₂=r₁₂, a₁₃=r₁₃P_(z)   (27)

a_(2l)=r₂₁, a₂₂=r₂₂, a₂₃=r₂₃P_(z)   (28)

 a ₃₁ =l _(x) , a ₃₂ =l _(y) , a ₃₃ =l _(z) P _(z) +s   (29)

Therefore, by substituting the expressions (27) to (29) for the expressions (17) to (26), it is possible to transform the expressions (17) to (26) into the following expressions (30) to (39). $\begin{matrix} {b_{11} = \frac{{{- l_{y}}r_{23}P_{z}} + {r_{22}\left( {{l_{z}P_{z}} + s} \right)}}{W_{1}}} & (30) \\ {b_{12} = \frac{{l_{y}r_{13}P_{z}} + {r_{12}\left( {{l_{z}P_{z}} + s} \right)}}{W_{1}}} & (31) \\ {b_{13} = \frac{{{- r_{22}}r_{23}P_{z}} + {r_{12}r_{23}P_{z}}}{W_{1}}} & (32) \\ {b_{21} = \frac{{l_{x}r_{23}P_{z}} - {r_{21}\left( {{l_{z}P_{z}} + s} \right)}}{W_{1}}} & (33) \\ {b_{22} = \frac{{{- l_{x}}r_{13}P_{z}} + {r_{11}\left( {{l_{z}P_{z}} + s} \right)}}{W_{1}}} & (34) \\ {b_{23} = \frac{{r_{21}r_{13}P_{z}} - {r_{11}r_{23}P_{z}}}{W_{1}}} & (35) \\ {b_{31} = \frac{{{- r_{22}}l_{x}} + {r_{21}l_{y}}}{W_{1}}} & (36) \\ {b_{32} = \frac{{r_{12}l_{x}} - {r_{11}l_{y}}}{W_{1}}} & (37) \\ {b_{33} = \frac{{{- r_{12}}r_{21}} + {r_{11}r_{22}}}{W_{1}}} & (38) \\ \begin{matrix} {W_{1} = \quad {{{- r_{22}}l_{x}r_{13}P_{z}} + {r_{21}l_{y}r_{13}P_{z}} + {r_{12}l_{x}r_{23}P_{z}} -}} \\ {\quad {{r_{11}l_{y}r_{23}P_{z}} - {r_{12}{r_{21}\left( {{l_{z}P_{z}} + s} \right)}} +}} \\ {\quad {r_{11}{r_{22}\left( {{l_{z}P_{z}} + s} \right)}}} \end{matrix} & (39) \end{matrix}$

Thus, by substituting the expressions (30) to (39) for the expressions (15) and (16), it is possible to obtain the read address (X_(M), Y_(M)) of the frame memory 11A in accordance with the following expressions (40) and (41). $\begin{matrix} \begin{matrix} {X_{M} = \quad {\frac{1}{H_{0}}\left\lbrack {{\left\{ {{{- l_{x}}r_{23}P_{z}} + {r_{22}\left( {{l_{z}P_{z}} + s} \right)}} \right\} X_{S}} +} \right.}} \\ {\quad {{\left\{ {{l_{y}r_{13}P_{z}} + {r_{12}\left( {{l_{z}P_{z}} + s} \right)}} \right\} Y_{S}} +}} \\ \left. \quad \left( {{{- r_{22}}r_{13}P_{z}} + {r_{12}r_{23}P_{z}}} \right) \right\rbrack \end{matrix} & (40) \\ \begin{matrix} {Y_{M} = \quad {\frac{1}{H_{0}}\left\lbrack {{\left\{ {{l_{x}r_{23}P_{z}} - {r_{21}\left( {{l_{z}P_{z}} + s} \right)}} \right\} X_{S}} +} \right.}} \\ {\quad {{\left\{ {{- l_{y}} + {r_{11}\left( {{l_{z}P_{z}} + s} \right)}} \right\} Y_{S}} +}} \\ \left. \quad \left( {{r_{21}r_{13}P_{z}} - {r_{11}r_{23}P_{z}}} \right) \right\rbrack \end{matrix} & (41) \end{matrix}$

The parameter H₀ is obtained from the following expression (42) in accordance with the expression (12).

H ₀=(−r ₂₂ l _(x) +r ₂₁ l _(y))X _(S)+(r ₁₂ l _(x) −r ₁₁ l _(y))Y _(S)+(−r ₁₂ r ₂₁ +r ₁₁ r ₂₂)  (42)

Thus, it is possible to show the read address (X_(M), Y_(M)) of the frame memory 11A by using parameters (r₁₁ to r₃₃, l_(x), l_(y), l_(z), s, and P_(Z)) of the transformation matrix T₃₃. Therefore, by supplying the screen address (X_(S), Y_(S)) from the expression (40) to the expression (42) so as to correspond to the luster scan sequence of the monitor screen 16, it is possible to successively obtain the read address (X_(M), Y_(M)) on the frame memory 11A corresponding to the supplied screen address.

Thus, the special effect system 10 of the present invention generates the read address (X_(M), Y_(M)) of the frame memory 11A (and the frame memory 11B) in accordance with the theory described above. Moreover, the special effect system 10 of the present invention obtains the matrix parameters b₁₁ to b₃₃ of the transformation matrix T₃₃ ⁻¹ by performing the operation of the expression (39) from the above expression (30) in accordance with the special effect parameters (r₁₁ to r₃₃, l_(x), l_(y), l_(z), s, and P_(Z)) input through the control panel 14 and supplies the matrix parameters to the read address generation circuit 11C to generate the read address (X_(M), Y_(M)) in accordance with the expressions (15) and (16). Thereby, it is possible to generate the video signal V₃ (that is, main image) to which the image transformation specified by an operator is applied.

(3) Symmetric image generation algorithm

In this section, an algorithm for generating a symmetric image point-symmetric, line-symmetric, or plane-symmetric to the main image generated by the above image transformation.

It is possible to easily obtain a symmetric image for a main image by making a symmetric transformation matrix work on the main image basically similarly to the case of the image transformation in the above-described three-dimensional space. That is, it is possible to easily obtain a symmetric image by further making a symmetric transformation matrix work on the transformation matrix T used to obtain a main image and making a resultingly-obtained transformation matrix work on the source video signal V₂. The symmetric transformation matrix is described in the subsequent sections every symmetric mode (point symmetry, line symmetry, or plane symmetry).

(3-1) Two-dimensional point symmetry

When a main image and a point serving as the symmetry yardstick are present on the xy plane of the above world coordinate system, it is possible to easily generate a symmetric image point-symmetric to the main image by performing two-dimensional point-symmetric transformation.

In this case, the two-dimensional point-symmetric transformation can be easily performed by the processing described below in general. That is, to obtain a point A′ based on a reference point p (α, β), which is point-symmetric to an optional point A on the xy plane as shown in FIG. 8, the point A is translated so as to translate the reference point p to the origin (0, 0) to obtain a point A₁ as shown in FIG. 9. Then, as shown in FIG. 10, a point A₂ is obtained which is point symmetric to the point A, to the origin (0, 0) as the reference. Then, a point A₂ is translated by the processing reverse to the above processing so as to translate the origin (0, 0) to the reference point p to obtain the point A′ as shown in FIG. 11. Thus, it is possible to obtain the point A′ point-symmetric to the optional point A by performing a series of these types of processing as shown in the following expression (43). $\begin{matrix} {\left\lbrack A\rightarrow A^{\prime} \right\rbrack = \left\lbrack {A\overset{{Translation}\quad}{\rightarrow}{A_{1}\overset{{{Origin}\quad {symmetry}}\quad}{\rightarrow}{A_{2}\overset{{{Inverse}\quad {translation}}\quad}{\rightarrow}A^{\prime}}}} \right\rbrack} & (43) \end{matrix}$

These types of processing are specifically described below. First, as shown in FIG. 9, it is possible to obtain the point A, from the point A by assuming a transformation matrix for translation from the origin (0, 0) to the point p (α, β) as Lp and making the inverse matrix Lp⁻¹ of the transformation matrix work on the point is. The inverse matrix Lp⁻¹ of the transformation matrix Lp from the origin (0, 0) to the point p (α, β) is shown by the following expression (44) $\begin{matrix} {{Lp}^{- 1} = \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ {- \alpha} & {- \beta} & 0 & 1 \end{bmatrix}} & (44) \end{matrix}$

Then, as shown in FIG. 10, it is possible to obtain the point A₂ from the point A₁ by making a point-symmetric transformation matrix S₀₂ using the origin (0, 0) as the point-symmetric reference point work on the point A₁. The point-symmetric transformation matrix S₀₂ based on the origin (0, 0) is shown by the following expression (45). $\begin{matrix} {S_{02} = \begin{bmatrix} {- 1} & 0 & 0 & 0 \\ 0 & {- 1} & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix}} & (45) \end{matrix}$

Then, as shown in FIG. 11, it is possible to obtain the point A′ from the point A₂ by making the transformation matrix Lp of translation from the origin (0, 0) to the point p (α, β) work on the point A₂. The transformation matrix Lp is shown by the following expression (46). $\begin{matrix} {{Lp} = \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ \alpha & \beta & 0 & 1 \end{bmatrix}} & (46) \end{matrix}$

Therefore, it is possible to obtain the point A′ from the optional point A at a stroke by making a symmetric transformation matrix Sp obtained by multiplying these transformation matrixes Lp⁻¹, S₀₂, and Lp in accordance with the processing sequence work on the point A. $\begin{matrix} \begin{matrix} {{Sp} = {{Lp}^{- 1} \cdot S_{02} \cdot {Lp}}} \\ {= {\begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ {- \alpha} & {- \beta} & 0 & 1 \end{bmatrix} \cdot \begin{bmatrix} {- 1} & 0 & 0 & 0 \\ 0 & {- 1} & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix} \cdot \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ \alpha & \beta & 0 & 1 \end{bmatrix}}} \\ {= \begin{bmatrix} {- 1} & 0 & 0 & 0 \\ 0 & {- 1} & 0 & 0 \\ 0 & 0 & 1 & 0 \\ {2\alpha} & {2\beta} & 0 & 1 \end{bmatrix}} \end{matrix} & (47) \end{matrix}$

The symmetric transformation matrix Sp is a matrix for obtaining a symmetric image point-symmetric to a main image. Therefore, it is possible to obtain a symmetric image from the input source video signal V₂ by making the transformation matrix T used to obtain a main image from the source video signal V₁ and the symmetric transformation matrix Sp work on the source video signal V₂. That is, it is possible to obtain a symmetric image from the source video signal V₂ by making a four-row four-column symmetric transformation matrix T_(2p). shown in the following expression (48) work on the source video signal V₂. $\begin{matrix} \begin{matrix} {T_{2p} = {{T \cdot {Sp}} = {T_{0} \cdot P_{0} \cdot {Sp}}}} \\ {= {\begin{bmatrix} r_{11} & r_{12} & r_{13} & 0 \\ r_{21} & r_{22} & r_{23} & 0 \\ r_{31} & r_{32} & r_{33} & 0 \\ l_{x} & l_{y} & l_{z} & s \end{bmatrix} \cdot {\begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & P_{z} \\ 0 & 0 & 0 & 1 \end{bmatrix}\quad\begin{bmatrix} {- 1} & 0 & 0 & 0 \\ 0 & {- 1} & 0 & 0 \\ 0 & 0 & 1 & 0 \\ {2\alpha} & {2\beta} & 0 & 1 \end{bmatrix}}}} \\ {= \begin{bmatrix} {{2\alpha \quad r_{13}P_{z}} - r_{11}} & {{2\beta \quad r_{13}P_{z}} - r_{12}} & r_{13} & {r_{13}\quad P_{z}} \\ {{2\alpha \quad r_{23}P_{z}} - r_{21}} & {{2\beta \quad r_{23}P_{z}} - r_{22}} & r_{23} & {r_{23}\quad P_{z}} \\ {{2\alpha \quad r_{33}P_{z}} - r_{31}} & {{2\beta \quad r_{33}P_{z}} - r_{32}} & r_{33} & {r_{33}\quad P_{z}} \\ {{2\alpha \quad \left( {{l_{z}P_{z}} + s} \right)} - l_{x}} & {{2\beta \quad \left( {{l_{z}P_{z}} + s} \right)} - l_{y}} & l_{z} & {{l_{z}P_{z}} + s} \end{bmatrix}} \end{matrix} & (48) \end{matrix}$

The special effect system 10 of the present invention generates the video signal V₆ of a symmetric image from the source video signal V₂ by actually obtaining a read address to which the image transformation shown by the symmetric transformation matrix T_(2p) will be applied and reading the source video signal V₂ from the frame memory 12A in accordance with the read address instead of multiplying the source video signal V₂ by the symmetric transformation matrix T_(2p).

That is, the special effect system 10 of the present invention generates the video signal V₆ of a symmetric image by successively writing the source video signal V₂ in the frame memory 12A and reading the source video signal V₂ in accordance with a read address to which the image transformation shown by the symmetric transformation matrix T_(2p) will be applied.

In this case, the source video signal V₂ written in the frame memory 12A and the video signal V₆ read from the frame memory 12A of the special effect system 10 are two-dimensional video data and the frame memory 12A is a memory for storing two-dimensional data. Therefore, in the case of the operation of a read address used for the read operation from the frame memory 12A, a parameter for computing the z-axis-directional data in a three-dimensional space is not practically used. Therefore, the parameter of the third column and third row for computing the z-axis-directional data is unnecessary.

That is, when assuming a symmetric transformation matrix actually required for the operation of a read address as T_(2p(33)), the symmetric transformation matrix T_(2p(33)) becomes a three-row three-column matrix excluding the third column and third row of the expression (48) and it is shown by the following expression (49). $\begin{matrix} {T_{2{p{(33)}}} = \begin{bmatrix} {{2\alpha \quad r_{13}P_{z}} - r_{11}} & {{2\beta \quad r_{13}P_{z}} - r_{12}} & {r_{13}\quad P_{z}} \\ {{2\alpha \quad r_{23}P_{z}} - r_{21}} & {{2\beta \quad r_{23}P_{z}} - r_{22}} & {r_{23}\quad P_{z}} \\ {{2\alpha \quad \left( {{l_{z}P_{z}} + s} \right)} - l_{x}} & {{2\beta \quad \left( {{l_{z}P_{z}} + s} \right)} - l_{y}} & {{l_{z}P_{z}} + s} \end{bmatrix}} & (49) \end{matrix}$

Moreover, in the case of the special effect system 10 of the present invention, it is necessary to obtain a point on the frame memory 12A corresponding to a point on the monitor screen 16 instead of obtaining a point on the monitor screen 16 corresponding to a point on the frame memory 12A as described in the section of the read-address generation method for the above-described three-dimensional image transformation. Therefore, the special effect system 10 of the present invention requires not the symmetric transformation matrix T_(2p(33)) but the inverse matrix T_(2p(33)) ⁻¹ of the symmetric transformation matrix T_(2p(33)).

In this case, by setting the parameters of the symmetric transformation matrix T_(2p(33)) as shown in the following expression (50), the inverse matrix T_(2p(33)) is shown by the following expression (51). $\begin{matrix} {T_{2{p{(33)}}} = \begin{bmatrix} a_{11}^{\prime} & a_{12}^{\prime} & a_{13}^{\prime} \\ a_{21}^{\prime} & a_{22}^{\prime} & a_{23}^{\prime} \\ a_{31}^{\prime} & a_{32}^{\prime} & a_{33}^{\prime} \end{bmatrix}} & (50) \\ {T_{2{p{(33)}}}^{- 1} = \begin{bmatrix} b_{11}^{\prime} & b_{12}^{\prime} & b_{13}^{\prime} \\ b_{21}^{\prime} & b_{22}^{\prime} & b_{23}^{\prime} \\ b_{31}^{\prime} & b_{32}^{\prime} & b_{33}^{\prime} \end{bmatrix}} & (51) \\ {{Where}{b_{ij} = \frac{a_{ji}}{\det \left( T_{2{p{(33)}}} \right)}}} & \quad \end{matrix}$

Thus, in the case of the special effect system 10, when a two-dimensional point symmetry is designated by an operator as the symmetric mode of a symmetric image, the CPU 13 obtains the parameters b₁₁′ to b₃₃′ of the inverse matrix T_(2p(33)) ⁻¹ in accordance with the designated reference point and supplies the parameters b₁₁′ to b₃₃′ to the read address generation circuit 12C. The read address generation circuit 12C calculates the read address (X_(M)′, Y_(M)′) of the frame memory 12A by using the parameters b₁₁′ to b₃₃′ for the symmetric transformation instead of the parameters b₁₁ to b₃₃ of the operation expressions shown in the expressions (15) and (16). Thus, the special effect system 10 can obtain the video signal V6 of a symmetric image two-dimensionally point-symmetric to a main image by reading the source video signal V₂ from the frame memory 12A in accordance with the read address (X_(M)′, Y_(M)′)

(3-2) Three-dimensional point symmetry

Next, in this section, three-dimensional point-symmetry is described. When a main image and a point serving as a symmetry yardstick are present in a three-dimensional space of the above-described world coordinate system, it is possible to easily generate a symmetric image point-symmetric to the main image by performing three-dimensional point-symmetric transformation.

In this case, the three-dimensional point-symmetric transformation can be easily performed by the processing described below in general. That is, to obtain a point A′ based on a reference point q (α, β, γ), which is point-symmetric to an optional point A in a three dimensional space as shown in FIG. 12, the point A is first translated so as to translate the reference point q (α, β, γ) to the origin (0, 0, 0) to obtain a point A₁ as shown in FIG. 13. Then, as shown in FIG. 14, a point-symmetric point A₂ based on the origin is obtained for the point A. Then, as shown in FIG. 15, the point A′ is obtained by translating the point A₂ in accordance with the processing reverse to the above processing to obtain the point A′ so as to translate the origin (0, 0, 0) to the reference point q. Thus, as shown in the following expression (52), it is possible to the point A′ point-symmetric to the optional point A by performing a series of these types of processing. $\begin{matrix} {\left\lbrack A\rightarrow A^{\prime} \right\rbrack = \left\lbrack {A\overset{Translation}{\rightarrow}{A_{1}\overset{\begin{matrix} {Origin} \\ {symmetry} \end{matrix}}{\rightarrow}{A_{2}\overset{\begin{matrix} {Inverse} \\ {translation} \end{matrix}}{\rightarrow}A^{\prime}}}} \right.} & (52) \end{matrix}$

These types of processing are specifically described below. First, to obtain a point A₁ from a point A as shown in FIG. 13, by assuming a transformation matrix of translation from the origin (0, 0, 0) to a point q (α, β, γ) as Lq, it is possible to obtain the point A₁ by making the inverse matrix Lq⁻¹ of the transformation matrix Lq work on the point A. The inverse matrix Lq⁻¹ of the transformation matrix Lq from the origin (0, 0, 0) to the point q (α, β, γ) is shown by the following expression (53). $\begin{matrix} {{Lq}^{- 1} = \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ {- \alpha} & {- \beta} & {- \gamma} & 1 \end{bmatrix}} & (53) \end{matrix}$

Then, as shown in FIG. 14, it is possible to obtain a point A₂ from the point A₁ by making a point-symmetric transformation matrix S₀₃ using the origin (0, 0, 0) as the reference point of point symmetry work on the point A₁. The point-symmetric transformation matrix S₀₃ based on the origin (0, 0, 0) is shown by the following expression (54). $\begin{matrix} {S_{03} = \begin{bmatrix} {- 1} & 0 & 0 & 0 \\ 0 & {- 1} & 0 & 0 \\ 0 & 0 & {- 1} & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix}} & (54) \end{matrix}$

Then, it is possible to obtain a point A′ from the point A₂ as shown in FIG. 15 by making the transformation matrix Lq of translation from the origin (0, 0, 0) to the point q (α, β, γ) work on the point A₂. The transformation matrix Lq is shown by the following expression (55). $\begin{matrix} {{Lq} = \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ \alpha & \beta & \gamma & 1 \end{bmatrix}} & (55) \end{matrix}$

Therefore, it is possible to obtain the point A′ point-symmetric to the optional point A from the point A at a stroke by making a symmetric transformation matrix Sq obtained by multiplying these transformation matrixes Lq⁻¹, S₀₃, and Lq in accordance with the processing sequence work on the point A as shown by the following expression (56). $\begin{matrix} \begin{matrix} {{Sq} = {{Lq}^{- 1} \cdot S_{03} \cdot {Lq}}} \\ {= {\begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ {- \alpha} & {- \beta} & {- \gamma} & 1 \end{bmatrix} \cdot \begin{bmatrix} {- 1} & 0 & 0 & 0 \\ 0 & {- 1} & 0 & 0 \\ 0 & 0 & {- 1} & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix} \cdot \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ \alpha & \beta & \gamma & 1 \end{bmatrix}}} \\ {= \begin{bmatrix} {- 1} & 0 & 0 & 0 \\ 0 & {- 1} & 0 & 0 \\ 0 & 0 & {- 1} & 0 \\ {2\alpha} & {2\beta} & {2\gamma} & 1 \end{bmatrix}} \end{matrix} & (56) \end{matrix}$

The symmetric transformation matrix Sq is a matrix for obtaining symmetric image point-symmetric to a main image. Therefore it is possible to obtain a symmetric image from the source video signal V₂ by making the transformation matrix T (comprising T₀ and P₀) used to obtain a main image from the source video signal V₁ and the symmetric transformation matrix Sq work on the source video signal V₂. That is, it is possible to obtain a symmetric image from the source video signal V₂ by making a four-row four-column symmetric transformation matrix T_(3p) shown by the following expression (57) work the source video signal V₂. $\begin{matrix} \begin{matrix} {T_{3p} = {T_{0} \cdot {Sq} \cdot P_{0}}} \\ {= {\begin{bmatrix} r_{11} & r_{12} & r_{13} & 0 \\ r_{21} & r_{22} & r_{23} & 0 \\ r_{31} & r_{32} & r_{33} & 0 \\ l_{x} & l_{y} & l_{z} & s \end{bmatrix} \cdot \begin{bmatrix} {- 1} & 0 & 0 & 0 \\ 0 & {- 1} & 0 & 0 \\ 0 & 0 & {- 1} & 0 \\ {2\alpha} & {2\beta} & {2\gamma} & 1 \end{bmatrix} \cdot \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & P_{z} \\ 0 & 0 & 0 & 1 \end{bmatrix}}} \\ {= \begin{bmatrix} {- r_{11}} & {- r_{12}} & 0 & {{- r_{13}}\quad P_{z}} \\ {- r_{21}} & {- r_{22}} & 0 & {{- r_{23}}\quad P_{z}} \\ {- r_{31}} & {- r_{32}} & 0 & {{- r_{33}}\quad P_{z}} \\ {{2\alpha \quad s} - l_{x}} & {{2\beta \quad s} - l_{y}} & 0 & {{\left( {{2\gamma \quad s} - l_{z}} \right)\quad P_{z}} + s} \end{bmatrix}} \end{matrix} & (57) \end{matrix}$

In this connection, though the symmetric transformation matrix Sp is made to work after the three-dimensional image transformation matrix T in the case of two dimensions, the symmetric transformation matrix Sq is inserted between the three-dimensional image transformation matrix T in the case of three dimensions. This is because it is necessary to perform symmetric transformation while a video signal is present in a three-dimensional space in the case of three dimensions.

Also in the case of three-dimensional point symmetry, the source video signal V₂ and video signal V₆ to be handled are two-dimensional data similarly to the case of two-dimensional point symmetry. Therefore, the parameter for computing z-axis-directional data of the three-dimensional symmetric transformation matrix T_(3p) is not practically used. Therefore, the parameter at the third column and third row for computing the z-axis-directional data of the symmetric transformation matrix T_(3p) shown by the expression (57) is unnecessary.

That is, when assuming a symmetric transformation matrix actually required for the operation of a read address as T_(3p(33)), the symmetric transformation matrix T_(3p(33)) becomes a three-row three-column matrix excluding the third column and third row of the expression (57) and it is shown by the following expression (58). $\begin{matrix} {T_{3{p{(33)}}} = \begin{bmatrix} {- r_{11}} & {- r_{12}} & {{- r_{13}}\quad P_{z}} \\ {- r_{21}} & {- r_{22}} & {{- r_{23}}\quad P_{z}} \\ {{2\alpha \quad s} - l_{x}} & {{2\beta \quad s} - l_{y}} & {{\left( {{2\gamma \quad s} - l_{z}} \right)\quad P_{z}} + s} \end{bmatrix}} & (58) \end{matrix}$

Moreover, because image transformation is performed in accordance with the read processing from the frame memory 12A similarly to the case of two-dimensional point symmetry in the case of three-dimensional point symmetry, it is necessary to obtain a point on the frame memory 12A corresponding to a point on the monitor screen 16. Therefore, the symmetric transformation matrix T_(3p(33)) is not actually necessary but the inverse matrix T_(3p(33)) ⁻¹ is necessary of the symmetric transformation matrix T_(3p(33)).

In this case, by setting the parameters of the symmetric transformation matrix T_(3p(33)) as shown in the following expression (59), the inverse matrix T_(3p(33)) ⁻¹ of the symmetric transformation matrix T_(3p(33)) is shown by the following expression (60). $\begin{matrix} {T_{3{p{(33)}}} = \begin{bmatrix} a_{11}^{\prime} & a_{12}^{\prime} & a_{13}^{\prime} \\ a_{21}^{\prime} & a_{22}^{\prime} & a_{23}^{\prime} \\ a_{31}^{\prime} & a_{32}^{\prime} & a_{33}^{\prime} \end{bmatrix}} & (59) \\ {T_{3{p{(33)}}}^{- 1} = \begin{bmatrix} b_{11}^{\prime} & b_{12}^{\prime} & b_{13}^{\prime} \\ b_{21}^{\prime} & b_{22}^{\prime} & b_{23}^{\prime} \\ b_{31}^{\prime} & b_{32}^{\prime} & b_{33}^{\prime} \end{bmatrix}} & (60) \\ {{Where}{b_{ij} = \frac{a_{ji}}{\det \left( T_{3{p{(33)}}} \right)}}} & \quad \end{matrix}$

Thus, in the case of the special effect system 10, when three-dimensional point symmetry is designated by an operator as the symmetric mode of a symmetric image, the CPU 13 obtains the parameters b₁₁′ to b₃₃′ of the inverse matrix T_(3p(33)) ⁻¹ in accordance with a designated reference point and supplies the parameters b₁₁′ to b₃₃′ to the read address generation circuit 12C. The read address generation circuit 12C calculates the read address (X_(M)′ Y_(M)′) of the frame memory 12A by using the parameters b₁₁′ to b₃₃′ instead of the parameters b₁₁ to b₃₃ of the operation expressions shown by the expressions (15) and (16). Thus, the special effect system 10 can obtain the video signal V₆ of a symmetric image three-dimensionally point-symmetric to a main image by reading the source video signal V₂ from the frame memory 12A in accordance with the read address (X_(M)′ Y_(M)′).

(3—3) Two-dimensional line symmetry

In this section, two-dimensional line symmetry is described. When a main image and a line serving as a symmetry yardstick are present on the xy plane of a world coordinate system, it is possible to easily generate a symmetric image line-symmetric to the main image by performing two-dimensional line-symmetric transformation.

In this case, the two-dimensional line-symmetric transformation can be easily performed by the processing to be described below in general. That is, to obtain a point A′ based on a line 1 passing through a point p (α, β) and forming an angle px with x-axis, which is line-symmetric to an optional point A on the xy plane as shown in FIG. 16, a point A₁ is obtained by translating the point A so as to translate the line 1 so that the point p (α, β) is superposed on the origin (0, 0) as shown in FIG. 17. Then, as shown in FIG. 18, a point A₂ is obtained by rotating the point A₁ clockwise about the origin by the angle θx formed between a line 1p obtained by translating the line 1 so as to pass through the origin and x-axis (that is, by rotating the point A₁ by −θx). Then, as shown in FIG. 19, a point A line-symmetric to the point A₂ is obtained on the basis of x-axis. Then, as shown in FIG. 20, a point A₄ is obtained by rotating the point A₃ counterclockwise about the origin by the angle θx (that is, by rotating the point A₃ by +θx inversely to the above case). Then, as shown in FIG. 21, a point A′ is obtained by translating the point A₄ through the processing reverse to the above case so as to translate the line 1p so that the origin (0, 0) is superposed on the point p. Thus, as shown in the following expression (61), it is possible to obtain the point A′ line-symmetric to the optional point A by performing a series of these types of processing. $\begin{matrix} {\left\lbrack A\rightarrow A^{\prime} \right\rbrack = \left\lbrack {A\overset{Translation}{\rightarrow}{A_{1}\overset{Rotation}{\rightarrow}{A_{2}\overset{\begin{matrix} \text{x-axis} \\ {symmetry} \end{matrix}}{\rightarrow}{A_{3}\overset{\begin{matrix} {Reverse} \\ {rotation} \end{matrix}}{\rightarrow}{A_{4}\overset{\begin{matrix} {Inverse} \\ {translation} \end{matrix}}{\rightarrow}A^{\prime}}}}}} \right\rbrack} & (61) \end{matrix}$

These types of processing are specifically described below. First, as shown in FIG. 17, the point A₁ can be obtained from the point A by assuming the transformation matrix of translation from the origin (0, 0) to the point p (α, β) as Lp and making the inverse matrix Lp⁻¹ of the transformation matrix work on the point A. The inverse matrix Lp⁻¹ of the transformation matrix from the origin. (0, 0) to the point p (α, β) is shown by the following expression (62) $\begin{matrix} {{Lp}^{- 1} = \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ {- \alpha} & {- \beta} & 0 & 1 \end{bmatrix}} & (62) \end{matrix}$

Then, as shown in FIG. 18, the point A₂ can be obtained from the point A₁ by making a rotational transformation matrix R(−θx) for rotating the point A₁ about the origin (0, 0) by −θx work on the point A₁. The rotational transformation matrix R(−θx) about the origin (0, 0) is shown by the following expression (63). $\begin{matrix} {{R\left( {{- \theta}\quad x} \right)} = \begin{bmatrix} {\cos \quad \left( {{- \theta}\quad x} \right)} & {\sin \quad \left( {{- \theta}\quad x} \right)} & 0 & 0 \\ {{- \sin}\quad \left( {{- \theta}\quad x} \right)} & {\cos \quad \left( {{- \theta}\quad x} \right)} & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix}} & (63) \end{matrix}$

Then, as shown in FIG. 19, the point A₃ can be obtained from the point A₂ by making a line-symmetric transformation matrix Sx based on x-axis work on the point A₂. The line-symmetric transformation matrix Sx based on x-axis is shown by the following expression (64). $\begin{matrix} {{Sx} = \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & {- 1} & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix}} & (64) \end{matrix}$

Then, as shown in FIG. 20, the point A₄ can be obtained from the point A₃ by making the rotational transformation matrix R(θx) for rotating the point A₃ by +θx about the origin (0, 0) work on the point A₃. The rotational transformation matrix R(θx) about the origin (0, 0) is shown by the following expression (65). $\begin{matrix} {{R\left( {\theta \quad x} \right)} = \begin{bmatrix} {\cos \quad \left( {\theta \quad x} \right)} & {\sin \quad \left( {\theta \quad x} \right)} & 0 & 0 \\ {{- \sin}\quad \left( {\theta \quad x} \right)} & {\cos \quad \left( {\theta \quad x} \right)} & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix}} & (65) \end{matrix}$

Then, as shown in FIG. 21, the point A′ can be obtained from the point A₄ by making the transformation matrix Lp of translation from the origin (0, 0) to the point p (α, β) work on the point A₄. The transformation matrix Lp is shown by the following expression (66). $\begin{matrix} {{Lp} = \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ \alpha & \beta & 0 & 1 \end{bmatrix}} & (66) \end{matrix}$

Therefore, the point A′ line-symmetric to an optional point A can be obtained from the point A at a stroke by making a symmetric transformation matrix S1 obtained by multiplying these transformation matrixes Lp⁻¹, R(−θx), Sx, R(θx), and Lp in accordance with the processing sequence as shown by the following expression (67). $\begin{matrix} \begin{matrix} {{Sl} = \quad {{Lp}^{- 1} \cdot {R\left( {{- \theta}\quad x} \right)} \cdot {Sx} \cdot {R\left( {\theta \quad x} \right)} \cdot {Lp}}} \\ {= \quad {\begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ {- \alpha} & {- \beta} & 0 & 1 \end{bmatrix} \cdot \begin{bmatrix} {\cos \quad \left( {{- \theta}\quad x} \right)} & {\sin \quad \left( {{- \theta}\quad x} \right)} & 0 & 0 \\ {{- \sin}\quad \left( {{- \theta}\quad x} \right)} & {\cos \quad \left( {{- \theta}\quad x} \right)} & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix} \cdot}} \\ {\quad {\begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & {- 1} & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix} \cdot \begin{bmatrix} {\cos \quad \left( {\theta \quad x} \right)} & {\sin \quad \left( {\theta \quad x} \right)} & 0 & 0 \\ {{- \sin}\quad \left( {\theta \quad x} \right)} & {\cos \quad \left( {\theta \quad x} \right)} & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix} \cdot}} \\ {\quad \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ \alpha & \beta & 0 & 1 \end{bmatrix}} \\ {= \quad \begin{bmatrix} {\cos \quad \left( {2\theta \quad x} \right)} & {\sin \quad \left( {2\theta \quad x} \right)} & 0 & 0 \\ {\sin \quad \left( {2\theta \quad x} \right)} & {{- \cos}\quad \left( {2\theta \quad x} \right)} & 0 & 0 \\ 0 & 0 & 1 & 0 \\ {K\quad \alpha} & {K\quad \beta} & 0 & 1 \end{bmatrix}} \end{matrix} & (67) \end{matrix}$

Where

Kα=α·(1·cos(2θx))−β·sin(2θx)

Kβ=−α·sin(2θx)+β·(1+cos(2θx))

The symmetric transformation matrix S1 is a matrix for obtaining a symmetric image line-symmetric to a main image. Therefore, the symmetric image can be obtained from the input source video signal V₂ by making the transformation matrix T used to obtain a main image from the source video signal V₁ and the symmetric transformation matrix S1 work on the source video signal V₂. That is by making the four-row four-column symmetric transformation matrix T₂₁ shown by the following expression (68) work on the source video signal V₂, the symmetric image can be obtained from the source video signal V₂. $\begin{matrix} \begin{matrix} {T_{21} = {T \cdot {Sl}}} \\ {= {T_{0} \cdot P_{0} \cdot {Sl}}} \\ {= {\begin{bmatrix} r_{11} & r_{12} & r_{13} & 0 \\ r_{21} & r_{22} & r_{23} & 0 \\ r_{31} & r_{32} & r_{33} & 0 \\ l_{x} & l_{y} & l_{z} & s \end{bmatrix} \cdot \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & P_{z} \\ 0 & 0 & 0 & 1 \end{bmatrix} \cdot \begin{bmatrix} {\cos \quad \left( {2\theta \quad x} \right)} & {\sin \quad \left( {2\theta \quad x} \right)} & 0 & 0 \\ {\sin \quad \left( {2\theta \quad x} \right)} & {{- \cos}\quad \left( {2\theta \quad x} \right)} & 0 & 0 \\ 0 & 0 & 1 & 0 \\ {K\quad \alpha} & {K\quad \beta} & 0 & 1 \end{bmatrix}}} \\ {= \begin{bmatrix} {{r_{11}\cos \quad \left( {2\theta \quad x} \right)} + {r_{12}{\sin \left( {2\theta \quad x} \right)}} + {K\quad \alpha \quad r_{13}P_{z}}} & {{r_{11}{\sin \left( {2\theta \quad x} \right)}} - {r_{12}\cos \quad \left( {2\theta \quad x} \right)} + {K\quad \beta \quad r_{13}P_{z}}} & r_{13} & {r_{13}P_{z}} \\ {{r_{21}\cos \quad \left( {2\theta \quad x} \right)} + {r_{22}{\sin \left( {2\theta \quad x} \right)}} + {K\quad \alpha \quad r_{23}P_{z}}} & {{r_{21}{\sin \left( {2\theta \quad x} \right)}} - {r_{22}\cos \quad \left( {2\theta \quad x} \right)} + {K\quad \beta \quad r_{23}P_{z}}} & r_{23} & {r_{23}P_{z}} \\ {{r_{31}\cos \quad \left( {2\theta \quad x} \right)} + {r_{32}{\sin \left( {2\theta \quad x} \right)}} + {K\quad \alpha \quad r_{33}P_{z}}} & {{r_{31}{\sin \left( {2\theta \quad x} \right)}} - {r_{32}\cos \quad \left( {2\theta \quad x} \right)} + {K\quad \beta \quad r_{33}P_{z}}} & r_{33} & {r_{33}P_{z}} \\ {{l_{x}\cos \quad \left( {2\theta \quad x} \right)} + {l_{y}{\sin \left( {2\theta \quad x} \right)}} + {K\quad \alpha \quad \left( {{l_{z}P_{z}} + s} \right)}} & {{l_{x}{\sin \left( {2\theta \quad x} \right)}} - {l_{y}\cos \quad \left( {2\theta \quad x} \right)} + {K\quad \beta \quad \left( {{l_{z}P_{z}} + s} \right)}} & l_{x} & {{l_{z}P_{z}} + s} \end{bmatrix}} \end{matrix} & (68) \end{matrix}$

Where

Kα=α·(1−cos(2θx))−β·sin(2θx)

Kβ=−αsin(2θx)+β·(1+cos(2θx))

Also in the case of two-dimensional line symmetry, the source video signal V₂ and video signal V₆ to be handled are two-dimensional video data similarly to the case of two-dimensional point symmetry. Therefore, a parameter for computing the z-axis-directional data of the symmetric transformation matrix T₂₁ is not practically used. Therefore, the parameter of the third column and third row for computing the z-axis-directional data of the symmetric transformation matrix T₂₁ shown by the expression (68) is unnecessary.

That is, when assuming a symmetric transformation matrix actually required for the operation of a read address as T₂₁₍₃₃₎, the symmetric transformation matrix T₂₁₍₃₃₎ becomes a three-row three-column matrix excluding the third column and third row of the expression (68) and it is shown by the following expression (69). $\begin{matrix} {T_{21{(33)}} = \begin{bmatrix} {{r_{11}\cos \quad \left( {2\theta \quad x} \right)} + {r_{12}{\sin \left( {2\theta \quad x} \right)}} + {K\quad \alpha \quad r_{13}P_{z}}} & {{r_{11}{\sin \left( {2\theta \quad x} \right)}} - {r_{12}\cos \quad \left( {2\theta \quad x} \right)} + {K\quad \beta \quad r_{13}P_{z}}} & {r_{13}P_{z}} \\ {{r_{21}\cos \quad \left( {2\theta \quad x} \right)} + {r_{22}{\sin \left( {2\theta \quad x} \right)}} + {K\quad \alpha \quad r_{23}P_{z}}} & {{r_{21}{\sin \left( {2\theta \quad x} \right)}} - {r_{22}\cos \quad \left( {2\theta \quad x} \right)} + {K\quad \beta \quad r_{23}P_{z}}} & {r_{23}P_{z}} \\ {{l_{x}\cos \quad \left( {2\theta \quad x} \right)} + {l_{y}{\sin \left( {2\theta \quad x} \right)}} + {K\quad \alpha \quad \left( {{l_{z}P_{z}} + s} \right)}} & {{l_{x}{\sin \left( {2\theta \quad x} \right)}} - {l_{y}\cos \quad \left( {2\theta \quad x} \right)} + {K\quad \beta \quad \left( {{l_{z}P_{z}} + s} \right)}} & {{l_{z}P_{z}} + s} \end{bmatrix}} & (69) \end{matrix}$

Where

Kα=α·(1−cos(2θx))−β·sin(2θx)

Kβ=−α·sin(2θx)+β·(1+cos(2θx))

Moreover, in the case of two-dimensional line symmetry, image transformation is performed by the read processing from the frame memory 12A similarly to the case of two-dimensional point symmetry. Therefore, it is necessary to obtain a point on the frame memory 12A corresponding to a point on the monitor screen 16. Therefore, in fact, the inverse matrix T₂₁₍₃₃₎ ⁻¹ of the symmetric transformation matrix T₂₁₍₃₃₎ is necessary though the symmetric transformation matrix T₂₁₍₃₃₎ is not necessary.

In this case, by setting the parameters of the symmetric transformation matrix T₂₁₍₃₃₎ as shown in the expression (70), the inverse matrix T₂₁₍₃₃₎ ⁻¹ is shown by the following expression (71). $\begin{matrix} {T_{21{(33)}} = \begin{bmatrix} a_{11}^{\prime} & a_{12}^{\prime} & a_{13}^{\prime} \\ a_{21}^{\prime} & a_{22}^{\prime} & a_{23}^{\prime} \\ a_{31}^{\prime} & a_{32}^{\prime} & a_{33}^{\prime} \end{bmatrix}} & (70) \\ {T_{21{(33)}}^{- 1} = \begin{bmatrix} b_{11}^{\prime} & b_{12}^{\prime} & b_{13}^{\prime} \\ b_{21}^{\prime} & b_{22}^{\prime} & b_{23}^{\prime} \\ b_{31}^{\prime} & b_{32}^{\prime} & b_{33}^{\prime} \end{bmatrix}} & (71) \\ {{Where}{b_{ij} = \frac{a_{ji}}{\det \left( T_{21{(33)}} \right)}}} & \quad \end{matrix}$

Thus, in the case of the special effect system 10, when two-dimensional line symmetry is designated by an operator as the symmetric mode of a symmetric image, the CPU 13 obtains the parameters b₁₁′ to b₃₃′ of the inverse matrix T₂₁₍₃₃₎ ⁻¹ in accordance with a designated reference line and supplies the parameters b₁₁′ to b₃₃′ to the read address generation circuit 12C. The read address generation circuit 12C calculates the read address (X_(M)′, Y_(M)′) of the frame memory 12A by using the parameters b₁₁′ to b₃₃′ for the symmetric transformation instead of the parameters b₁₁ to b₃₃ for the operation expressions shown in the above-described expressions (15) and (16). Thus, by reading the source video signal V₂ from the frame memory 12A in accordance with the read address (X_(M)′, Y_(M)′), the special effect system 10 can obtain the video signal V₆ of a symmetric image two-dimensionally line-symmetric to a main image.

(3-4) Three-dimensional line symmetry

In this section, three-dimensional line symmetry is described. When a main image and a line serving as a symmetry yardstick are present in a three-dimensional space, it is possible to easily generate a symmetric image line-symmetric to the main image by performing three-dimensional line-symmetric transformation.

In this case, three-dimensional line-symmetric transformation can be easily performed in accordance with the processing described below in general. That is, to obtain a point A′ based on a line m passing through a point q (α, β, γ) and forming angles θx, θy, and θz with x-axis, y-axis, and z-axis, which is line-symmetric to an optional point A in a three-dimensional space as shown in FIG. 22, the point A is translated so as to translate the line m so that the point q (α, β, γ) is superposed on the origin (0, 0, 0) to obtain a point A₁ as shown in FIG. 23. Then, as shown in FIG. 24, a line-symmetric point A₂ based on a line 1 is obtained by translating the line m so as to pass through the origin. Then, as shown in FIG. 25, the point A′ is obtained by translating the point A₂ by the processing reverse to the above processing so as to translate the line 1 so that the origin (0, 0, 0) is superposed on the point q. Thus, as shown by the following expression (72), the point A′ line-symmetric to the optional point A can be obtained by performing the a series of the above types of processing. $\begin{matrix} {\left\lbrack A\rightarrow A^{\prime} \right\rbrack = \left\lbrack {A\overset{Translation}{\rightarrow}{A_{1}\overset{\begin{matrix} {Line} \\ {symmetry} \end{matrix}}{\rightarrow}{A_{2}\overset{\begin{matrix} {Inverse} \\ {translation} \end{matrix}}{\rightarrow}A^{\prime}}}} \right\rbrack} & (72) \end{matrix}$

Hereafter, these types of processing are specifically described. First, as shown in FIG. 23, the point A₁ can be obtained from the point A by assuming the transformation matrix of translation from the origin (0, 0, 0) to the point q (α, β, γ) as Lq and making the inverse matrix Lq⁻¹ of the transformation matrix Lq work on the point A. The inverse matrix Lq⁻¹ of the transformation matrix Lq from the origin (0, 0, 0) to the point q (α, β, γ) is shown by the following expression (73). $\begin{matrix} {{Lq}^{- 1} = \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ {- \alpha} & {- \beta} & {- \gamma} & 1 \end{bmatrix}} & (73) \end{matrix}$

Then, as shown in FIG. 24, the point A₂ can be obtained from the point A₁ about the line 1 obtained by translating the line m so as to pass through the origin and making a rotational transformation matrix S_(L) for rotating the point A₁ by an angle π work on the point A₁. In general, a rotational transformation matrix R1(θ) for rotating an object by an angle θ in a three-dimensional space about the line 1 passing through the origin and forming angles θx, θy, and θz with x-axis, y-axis, and z-axis becomes a four-row four-column matrix and it is shown by the following expression (74). $\begin{matrix} {{{Rl}(\theta)} = \begin{bmatrix} {a^{2} + {\left( {1 - a^{2}} \right)\quad \cos \quad (\theta)}} & {{a\quad b\quad \left( {1 - {\cos \quad \theta}} \right)} + {c\quad \sin \quad \theta}} & {{a\quad c\quad \left( {1 - {\cos \quad \theta}} \right)\quad b} - {\sin \quad \theta}} & 0 \\ {{a\quad b\quad \left( {1 - {\cos \quad \theta}} \right)} - {c\quad \sin \quad \theta}} & {b^{2} + {\left( {1 - b^{2}} \right)\quad \cos \quad (\theta)}} & {{b\quad c\quad \left( {1 - {\cos \quad \theta}} \right)} + {a\quad \sin \quad \theta}} & 0 \\ {{a\quad c\quad \left( {1 - {\cos \quad \theta}} \right)} + {b\quad \sin \quad \theta}} & {{b\quad c\quad \left( {1 - {\cos \quad \theta}} \right)} - {a\quad \sin \quad \theta}} & {c^{2} + {\left( {1 - c^{2}} \right)\quad \cos \quad (\theta)}} & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix}} & (74) \end{matrix}$

Where

a=cosθx, b=cosθy, c=cosθz

The rotational transformation matrix S_(L) is transformation processing for rotating an object by an angle π in a three-dimensional space about the line 1. Therefore, by substituting θ=π for the expression (74), the rotational transformation matrix S_(L) is shown by the following expression (75). $\begin{matrix} \begin{matrix} {S_{L} = {{Rl}(\pi)}} \\ {= \begin{bmatrix} {{2a^{2}} + 1} & {2a\quad b} & {2a\quad c} & 0 \\ {2a\quad b} & {{2b^{2}} + 1} & {2b\quad c} & 0 \\ {2a\quad c} & {2b\quad c} & {{2c^{2}} + 1} & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix}} \end{matrix} & (75) \end{matrix}$

Where

a=cosθx, b=cosθy, c=cosθz

Then, as shown in FIG. 25, the point A′ can be obtained from the point A₂ by making the transformation matrix Lq of translation from the origin (0, 0, 0) to the point q (α, β, γ) work on the point A₂. The transformation matrix Lq is shown by the following expression (76). $\begin{matrix} {{Lq} = \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ \alpha & \beta & \gamma & 1 \end{bmatrix}} & (76) \end{matrix}$

Therefore, the line-symmetric point A′ can be obtained from the point A at a stroke by making the symmetric transformation matrix S1 obtained by multiplying these transformation matrixes Lq⁻¹, S_(L), and Lq in accordance with the processing sequence work on the point A as shown by the following expression (77). $\begin{matrix} \begin{matrix} {{Sl} = \quad {{Lq}^{- 1} \cdot S_{L} \cdot {Lq}}} \\ {= \quad {\begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ {- \alpha} & {- \beta} & {- \gamma} & 1 \end{bmatrix} \cdot \begin{bmatrix} {{2a^{2}} + 1} & {2a\quad b} & {2a\quad c} & 0 \\ {2a\quad b} & {{2b^{2}} + 1} & {2b\quad c} & 0 \\ {2a\quad c} & {2b\quad c} & {{2c^{2}} + 1} & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix} \cdot}} \\ {\quad \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ \alpha & \beta & \gamma & 1 \end{bmatrix}} \\ {= \quad \begin{bmatrix} {{2a^{2}} + 1} & {2a\quad b} & {2a\quad c} & 0 \\ {2a\quad b} & {{2b^{2}} + 1} & {2b\quad c} & 0 \\ {2a\quad c} & {2b\quad c} & {{2c^{2}} + 1} & 0 \\ K_{x} & K_{y} & K_{z} & 1 \end{bmatrix}} \end{matrix} & (77) \end{matrix}$

Where

K _(x)=−2a(αa+βb+γc)

K _(y)=−2b(αa+βb+γc)

K _(z)=−2c(αa+βb+γc)

a=cos θx, b=cos θy, c=cos θz

The symmetric transformation matrix S1 is a matrix for obtaining a symmetric image line-symmetric to a main image. Therefore, a symmetric image can be obtained from the input source video signal V₂ by making the transformation matrix T (comprising T₀ and P₀) used to obtain a main image from the source video signal V₁ and the symmetric transformation matrix S1 work on the source video signal V₂. That is, a symmetric image can be obtained from the source video signal V₂ by making the four-row four-column symmetric transformation matrix T₃₁ shown by the following expression (78) work on the source video signal V₂. $\begin{matrix} \begin{matrix} {T_{31} = \quad {T_{0} \cdot {Sl} \cdot P_{0}}} \\ {= \quad {\begin{bmatrix} r_{11} & r_{12} & r_{13} & 0 \\ r_{21} & r_{22} & r_{23} & 0 \\ r_{31} & r_{32} & r_{33} & 0 \\ l_{x} & l_{y} & l_{z} & s \end{bmatrix} \cdot \begin{bmatrix} {{2a^{2}} + 1} & {2a\quad b} & {2a\quad c} & 0 \\ {2a\quad b} & {{2b^{2}} + 1} & {2b\quad c} & 0 \\ {2a\quad c} & {2b\quad c} & {{2c^{2}} + 1} & 0 \\ K_{x} & K_{y} & K_{z} & 1 \end{bmatrix} \cdot \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & P_{z} \\ 0 & 0 & 0 & 1 \end{bmatrix}}} \\ {= \quad \begin{bmatrix} {{r_{11}\left( {{2a^{2}} + 1} \right)} + {2a\quad b\quad r_{12}} + {2a\quad c\quad r_{13}}} & {{2a\quad b\quad r_{11}} + {r_{12}\left( {{2b^{2}} + 1} \right)} + {2b\quad c\quad r_{13}}} & 0 & {P_{z}\left\lbrack {{2a\quad c\quad r_{11}} + {2b\quad c\quad r_{12}} + {r_{13}\quad \left( {{2c^{2}} + 1} \right)}} \right\rbrack} \\ {{r_{21}\left( {{2a^{2}} + 1} \right)} + {2a\quad b\quad r_{22}} + {2a\quad c\quad r_{23}}} & {{2a\quad b\quad r_{21}} + {r_{22}\left( {{2b^{2}} + 1} \right)} + {2b\quad c\quad r_{23}}} & 0 & {P_{z}\left\lbrack {{2a\quad c\quad r_{21}} + {2b\quad c\quad r_{22}} + {r_{23}\quad \left( {{2c^{2}} + 1} \right)}} \right\rbrack} \\ {{r_{31}\left( {{2a^{2}} + 1} \right)} + {2a\quad b\quad r_{32}} + {2a\quad c\quad r_{33}}} & {{2a\quad b\quad r_{31}} + {r_{32}\left( {{2b^{2}} + 1} \right)} + {2b\quad c\quad r_{33}}} & 0 & {P_{z}\left\lbrack {{2a\quad c\quad r_{31}} + {2b\quad c\quad r_{32}} + {r_{33}\quad \left( {{2c^{2}} + 1} \right)}} \right\rbrack} \\ {{l_{x}\left( {{2a^{2}} + 1} \right)} + {2a\quad b\quad l_{y}} + {2a\quad c\quad l_{z}} + {K_{x}s}} & {{2a\quad b\quad l_{x}} + {l_{y}\left( {{2b^{2}} + 1} \right)} + {2b\quad c\quad l_{z}} + {K_{y}s}} & 0 & {P_{z}\left\lbrack {{2a\quad c\quad l_{x}} + {2b\quad c\quad l_{y}} + {l_{z}\quad \left( {{2c^{2}} + 1} \right)} + {K_{z}s}} \right\rbrack} \end{bmatrix}} \end{matrix} & (78) \end{matrix}$

Where

K _(x)=−2a(αa+βb+γc)

K _(y)=−2b(αa+βb+γc)

K _(z)=−2c(αa+βb+γc)

a=cos θx, b=cos θy, c=cos θz

In the case of three-dimensional line symmetry, the source video signal V₂ and video signal V₆ to be handled are two-dimensional data similarly to the case of two-dimensional line symmetry. Therefore, a parameter for computing z-axis-directional data of the symmetric transformation matrix T₃₁ is not practically used. Therefore, the parameter of third column and third row for computing z-axis-directional data of the symmetric transformation matrix T₃₁ shown by the expression (78) is unnecessary.

That is, when assuming a symmetric transformation matrix actually required for the operation of a read address as T₃₁₍₃₃₎, the symmetric transformation matrix T₃₁₍₃₃₎ becomes a three-row three-column matrix excluding the third column and third row of the expression (78) and it is shown by the following expression (79). $\begin{matrix} {T_{31{(33)}} = \begin{bmatrix} {{r_{11}\left( {{2a^{2}} + 1} \right)} + {2a\quad b\quad r_{12}} + {2a\quad c\quad r_{13}}} & {{2a\quad b\quad r_{11}} + {r_{12}\left( {{2b^{2}} + 1} \right)} + {2b\quad c\quad r_{13}}} & {P_{z}\left\lbrack {{2a\quad c\quad r_{11}} + {2b\quad c\quad r_{12}} + {r_{13}\quad \left( {{2c^{2}} + 1} \right)}} \right\rbrack} \\ {{r_{21}\left( {{2a^{2}} + 1} \right)} + {2a\quad b\quad r_{22}} + {2a\quad c\quad r_{23}}} & {{2a\quad b\quad r_{21}} + {r_{22}\left( {{2b^{2}} + 1} \right)} + {2b\quad c\quad r_{23}}} & {P_{z}\left\lbrack {{2a\quad c\quad r_{21}} + {2b\quad c\quad r_{22}} + {r_{23}\quad \left( {{2c^{2}} + 1} \right)}} \right\rbrack} \\ {{l_{x}\left( {{2a^{2}} + 1} \right)} + {2a\quad b\quad l_{y}} + {2a\quad c\quad l_{z}} + {K_{x}s}} & {{2a\quad b\quad l_{x}} + {l_{y}\left( {{2b^{2}} + 1} \right)} + {2b\quad c\quad l_{z}} + {K_{y}s}} & {P_{z}\left\lbrack {{2a\quad c\quad l_{x}} + {2b\quad c\quad l_{y}} + {l_{z}\quad \left( {{2c^{2}} + 1} \right)} + {K_{z}s}} \right\rbrack} \end{bmatrix}} & (79) \end{matrix}$

where

K _(x)=−2a(αa+βb+γc)

K _(y)=−2b(αa+βb+γc)

K _(z)=−2c(αa+βb+γc)

a=cos θx, b=cos θy, c=cos θz

Moreover, in the case of three-dimensional line symmetry, image transformation is performed in accordance with the read processing from the frame memory 12A similarly to the case of two-dimensional line symmetry. Therefore, it is necessary to obtain a point on the frame memory 12A corresponding to a point on the monitor screen 16. Therefore, in fact, the symmetric transformation matrix T₃₁₍₃₃₎ is not necessary but the inverse matrix T₃₁₍₃₃₎ ⁻¹ of the symmetric transformation matrix T₃₁₍₃₃₎ is necessary.

In this case, by setting the parameters of the symmetric transformation matrix T₃₁₍₃₃₎ as shown in the following expression (80), the inverse matrix T₃₁₍₃₃₎ ⁻¹ of the symmetric transformation matrix T₃₁₍₃₃₎ is shown by the following expression (81). $\begin{matrix} {T_{31{(33)}} = \begin{bmatrix} a_{11}^{\prime} & a_{12}^{\prime} & a_{13}^{\prime} \\ a_{21}^{\prime} & a_{22}^{\prime} & a_{23}^{\prime} \\ a_{31}^{\prime} & a_{32}^{\prime} & a_{33}^{\prime} \end{bmatrix}} & (80) \\ {T_{31{(33)}}^{- 1} = \begin{bmatrix} b_{11}^{\prime} & b_{12}^{\prime} & b_{13}^{\prime} \\ b_{21}^{\prime} & b_{22}^{\prime} & b_{23}^{\prime} \\ b_{31}^{\prime} & b_{32}^{\prime} & b_{33}^{\prime} \end{bmatrix}} & (81) \\ {{Where}{b_{ij} = \frac{a_{ji}}{\det \left( T_{31{(33)}} \right)}}} & \quad \end{matrix}$

Thus, in the case of the special effect system 10, when three-dimensional line symmetry is designated by an operator as the symmetric mode of a symmetric image, the CPU 13 obtains the parameters b₁₁′ to b₃₃′ of the inverse matrix T₃₁₍₃₃₎ ⁻¹ in accordance with a designated reference line and supplies the parameters b₁₁′ to b₃₃′ to the read address generation circuit 12C. The read address generation circuit 12C calculates the read address (X_(M)′, Y_(M)′) of the frame memory 12A by using the parameters b₁₁′ to b₃₃′ for the symmetric transformation instead of the parameters b₁₁ to b₃₃ of the operation expressions shown in the expressions (15) and (16). Thus, by reading the source video signal V₂ from the frame memory 12A in accordance with the read address (X_(M)′, Y_(M)′), the special effect system 10 can obtain the video signal V₆ of a symmetric image three-dimensionally line-symmetric to a main image.

(3-5) Plane symmetry

In this section, plane symmetry is described. When a main image and a plane serving as a symmetry yardstick are present in a three-dimensional space, it is possible to easily generate a symmetric image plane-symmetric to the main image by performing plane symmetric transformation.

In this case, the plane symmetric transformation can be easily performed by the processing described below in general. That is, as shown in FIG. 26, to obtain a point A′ plane-symmetric to an optional point A in a three-dimensional space, which is based on a plane G passing through a point q (α, β, γ) and whose normal vector has angles θx, θy, and θz formed with x-axis, y-axis, and z-axis, a point A₁ is obtained by translating the point A so as to translate the plane G so that the point q (α, β, γ) is superposed on the origin (0, 0, 0). Then, as shown in FIG. 28, a point A₃ present in a three-dimensional space is obtained by applying the processing same as the processing for rotating a plane G₀ obtained by translating the plane G so as to pass through the origin so that the plane G₀ is superposed on the xy plane to the point A₁. Then, as shown in FIG. 28, a point A₄ plane-symmetric to the point A₃ is obtained on the basis of the xy plane. Then, as shown in FIG. 29, a point A₆ present in a three-dimensional space is obtained by applying to the point A₄ the rotation processing for returning the plane G₀ superposed on the xy plane to its original position. Then, the point A′ is obtained by translating the point A₆ so as to translate the plane G₀ so that the origin (0, 0, 0) is superposed on the point q (that is, so as to return the plane G₀ to the plane G).

In this connection, the processing for superposing the plane G₀ on the xy plane is executed by rotating the plane G₀ up to a predetermined angle about x-axis and thereafter rotating it up to a predetermined angle about y-axis. Therefore, the point A₃ is obtained from the point A₁ by applying the processing for rotating the point A₁ about x-axis to the point A₁ to obtain the point A₂ and thereafter, rotating the point A₂ about y-axis. Moreover, similarly, the processing for returning the plane G₀ superposed on the xy plane to its original position is executed by reversing the plane G₀ on the xy plane about y-axis and thereafter reversing the plane G₀ about x-axis. Therefore, the point A₆ is obtained from the point A₄ by first applying the processing for reversing the point A₄ about y-axis to the point A₄ to obtain a point A₅ about y-axis and thereafter reversing the point A₅ about x-axis.

Thus, the above types of processing are arranged as shown by the following expression (82). $\begin{matrix} {\left\lbrack A\rightarrow A^{\prime} \right\rbrack = \left\lbrack {A\overset{Translation}{\rightarrow}{A_{1}\overset{\begin{matrix} \text{x-axis} \\ {rotation} \end{matrix}}{\rightarrow}{A_{2}\overset{\begin{matrix} \text{y-axis} \\ {rotation} \end{matrix}}{\rightarrow}{A_{3}\overset{\begin{matrix} \text{xy-plane} \\ {symmetry} \end{matrix}}{\rightarrow}{A_{4}\overset{\begin{matrix} \text{y-axis} \\ {reversal} \end{matrix}}{\rightarrow}{A_{5}\overset{\begin{matrix} \text{x-axis} \\ {reversal} \end{matrix}}{\rightarrow}{A_{6}\overset{\begin{matrix} {Inverse} \\ {translation} \end{matrix}}{\rightarrow}A^{\prime}}}}}}}} \right\rbrack} & (82) \end{matrix}$

By performing a series of the types of processing shown by the expression (82), the point A′ plane-symmetric to the optional point A can be obtained.

These types of processing are specifically described below. First, as shown in FIG. 27, the point A₁ can be obtained from the point A by assuming a transformation matrix of translation from the origin (0, 0, 0) to the point q (α, β, γ) as Lq and making the inverse matrix Lq⁻¹ of the transformation matrix Lq work on the point A. The inverse matrix Lq⁻¹ of the transformation matrix Lq from the origin (0, 0, 0) to the point q (α, β, γ) is shown by the following expression (83). $\begin{matrix} {{Lq}^{- 1} = \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ {- \alpha} & {- \beta} & {- \gamma} & 1 \end{bmatrix}} & (83) \end{matrix}$

Next, the process for obtaining the point A₃ from the point A₁ is described below. First, in the processing until obtaining the point A₃, the rotation processing about x-axis and the rotation processing about y-axis are performed. A rotational transformation matrix Rx(δx) for the rotation processing about x-axis is shown by the following expression (84) by assuming a rotation angle δx and substituting a=1 and b=c=0 for the above expression (74). $\begin{matrix} {{{Rx}\left( {\delta \quad x} \right)} = \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & {\cos \quad \delta \quad x} & {\sin \quad \delta \quad x} & 0 \\ 0 & {{- \sin}\quad \delta \quad x} & {\cos \quad \delta \quad x} & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix}} & (84) \end{matrix}$

Moreover, a rotational transformation matrix Ry(δy) for the rotation processing about y-axis is shown by the following expression (85) by assuming a rotation angle as δy and substituting a=c=0 and b=1 for the expression (74). $\begin{matrix} {{{Ry}\left( {\delta \quad y} \right)} = \begin{bmatrix} {\cos \quad \delta \quad y} & 0 & {{- \sin}\quad \delta \quad y} & 0 \\ 0 & 1 & 0 & 0 \\ {\sin \quad \delta \quad y} & 0 & {\cos \quad \delta \quad y} & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix}} & (85) \end{matrix}$

The rotation angle for rotation about x-axis and y-axis is described below. First, it is assumed that the normal vector of the plane G is g=(a, b, c). When the normal vector of the plane G is g, the normal vector of the plane G₀ translated in a three-dimensional space has the same vector g. Because the normal vector g is a vector vertical to the planes G and G₀, it is possible to superpose the plane G₀ on the xy plane and obtain the point A₃ by rotating the plane G₀ about x-axis and y-axis so that the normal vector g becomes parallel with z-axis. Therefore, it is possible to determine a rotation angle in accordance with the normal vector g.

First, because the normal vector g forms angles θx, θy, and θz with x-axis, y-axis and z-axis, the relation shown by the following expressions (86) is obtained.

a=cosθx, b=cosθy, c=cosθz  (86)

As shown in FIG. 30, when assuming a vector obtained by projecting the normal vector g on the yz plane as gyz, the angle formed between the vector gyz and z-axis as θyz, and an angle formed between the vector gyz and vector g as θxz, the angles θyz and θxz are shown by the following expressions (87) and (88) respectively.

θyz=tan⁻¹(b/c)  (87)

θxz=tan⁻¹[a/{square root over ((b²+L +c²+L ))}]  (88)

Moreover, by substituting the value of the expression (86) for the expressions (87) and (88), the angles θyz and θxz are transformed as shown by the following expressions (89) and (90).

θyz=tan⁻¹(Cos θy/cos θz)  (89)

θxz=tan⁻¹[cosθx/{square root over ((cos²+L θy+cos²+L θz))}  (90)

By rotating the normal vector g about x-axis by the angle θyz obtained for the yz plane and moreover rotating it about y-axis by the angle θxz obtained for the xz plane, the normal vector g is transformed into a vector parallel with z-axis while keeping its magnitude. That is, by performing these types of processing for the plane G₀, the plane G₀ is superposed on a passing through the origin and with the normal vector parallel 18 z-axis, that is, the xy plane.

Therefore, by making the rotational transformation matrix Rx(−δx) obtained by substituting −δx comprising δx=θyz for the expression (84) work on the normal vector g as shown in the following expression (91), the component in the y-axis direction becomes “0” and the normal vector g is transformed into a vector g₂ on the xz plane as shown in FIG. 31. Therefore, by making the rotational transformation matrix Rx (−δx) work on the point A₁, it is possible to obtain the point A₂ from the point A₁. $\begin{matrix} {{{Rx}\left( {{- \delta}\quad x} \right)} = \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & {\cos \quad \delta \quad x} & {{- \sin}\quad \delta \quad x} & 0 \\ 0 & {\sin \quad \delta \quad x} & {\cos \quad \delta \quad x} & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix}} & (91) \end{matrix}$

Similarly, by making the rotational transformation matrix Ry(−δy) obtained by substituting −δy comprising δy=θxz for the expression (85) work on the vector g₂ as shown by the following expression (92), the component in the x-axis direction becomes “0” and the vector g₂ on the xz plane is transformed into a vector g₃ parallel with z-axis as shown in FIG. 32. Therefore, by making the rotational transformation matrix Ry(−δy) work on the point A₂, it is possible to obtain the point A₃ from the point A₂. $\begin{matrix} {{{Ry}\left( {{- \delta}\quad y} \right)} = \begin{bmatrix} {\cos \quad \delta \quad y} & 0 & {\sin \quad \delta \quad y} & 0 \\ 0 & 1 & 0 & 0 \\ {{- \sin}\quad \delta \quad y} & 0 & {\cos \quad \delta \quad y} & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix}} & (92) \end{matrix}$

Then, a point A₄ can be obtained from the point A₃ thus obtained by applying the plane-symmetric transformation related to the xy plane to the point A₃. In this case, the plane-symmetric transformation related to the xy plane can be performed by reversing the code of the value of the z coordinate. Therefore, the transformation matrix S_(xy) of the plane-symmetric transformation is shown by the following expression (93). $\begin{matrix} {S_{xy} = \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & {- 1} & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix}} & (93) \end{matrix}$

Then, a point A₅ can be obtained from the point A₄ obtained by making the plane-symmetric transformation matrix S_(xy) work on the point A₄ by rotating the point A₄ about y-axis by an angle (+δy) opposite to the case of the above rotation processing. That is, by making the rotational transformation matrix Ry(δy) obtained by substituting δy comprising δy=θxz for the expression (85) work as shown by the following expression (94), the vector g₃ parallel with z-axis is returned to the vector g₂ as shown in FIG. 33. Therefore, by making the rotational transformation matrix Ry(δy) work on the point A₄, it is possible to obtain the point A₅ from the point A₄. $\begin{matrix} {{{Ry}\left( {\delta \quad y} \right)} = \begin{bmatrix} {\cos \quad \delta \quad y} & 0 & {{- \sin}\quad \delta \quad y} & 0 \\ 0 & 1 & 0 & 0 \\ {\sin \quad \delta \quad y} & 0 & {\cos \quad \delta \quad y} & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix}} & (94) \end{matrix}$

where

δy=θxz=tan⁻¹[cosθx/{square root over ((cos²+L θy+cos²+L θz))}

Then, a point A₆ can be obtained from the point A₅ by rotating the point A₅ about x-axis by an angle (+δx) opposite to the case of the above rotation processing. That is, by making the rotational transformation matrix Rx(δx) obtained by substituting δx comprising δx=θyz for the expression (84) work as shown by the following expression (95), the vector g₂ on the xz plane is returned to the normal vector g as shown in FIG. 34. Therefore, by making the rotational transformation matrix Rx(δx) work on the point A₅, it is possible to obtain the point A₆ from the point A₅. $\begin{matrix} {{{Rx}\left( {\delta \quad x} \right)} = \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & {\cos \quad \delta \quad x} & {\sin \quad \delta \quad x} & 0 \\ 0 & {{- \sin}\quad \delta \quad x} & {\cos \quad \delta \quad x} & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix}} & (95) \end{matrix}$

where

δx=θyz=tan⁻¹(cos θy/cosθz)

Finally, a point A′ can be obtained from the point A₆ thus obtained by making the transformation matrix Lq for translation from the origin (0, 0, 0) to the point q (α, β, γ) work on the point A₆. The transformation matrix Lq is shown by the following expression (96). $\begin{matrix} {{Lq} = \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ \alpha & \beta & \gamma & 1 \end{bmatrix}} & (96) \end{matrix}$

Thus, the point A′ plane-symmetric to an optional pint A can be obtained from the point A at a stroke by making a symmetric transformation matrix Sg obtained by multiplying these transformation matrixes Lq⁻¹, Rx(−δx), Ry(−δy), S_(xy), Ry(δy), Rx(δx) and Lq in accordance with the processing sequence work on the point A. $\begin{matrix} \begin{matrix} {{Sg} = \quad {{{Lq}^{- 1} \cdot {{Rx}\left( {{- \delta}\quad x} \right)} \cdot {{Ry}\left( {{- \delta}\quad y} \right)} \cdot S_{xy} \cdot {Ry}}\quad {\left( {\delta \quad y} \right) \cdot {{Rx}\left( {\delta \quad x} \right)} \cdot {Lq}}}} \\ {= \quad {\begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ {- \alpha} & {- \beta} & {- \gamma} & 1 \end{bmatrix} \cdot \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & {\cos \quad \delta \quad x} & {{- \sin}\quad \delta \quad x} & 0 \\ 0 & {\sin \quad \delta \quad x} & {\cos \quad \delta \quad x} & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix} \cdot}} \\ {\quad {{{\begin{bmatrix} {\cos \quad \delta \quad y} & 0 & {{- \sin}\quad \delta \quad y} & 0 \\ 0 & 1 & 0 & 0 \\ {\sin \quad \delta \quad y} & \quad & {\cos \quad \delta \quad y} & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix}\quad\begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & {- 1} & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix}}\quad\begin{bmatrix} {\cos \quad \delta \quad y} & 0 & {{- \sin}\quad \delta \quad y} & 0 \\ 0 & 1 & 0 & 0 \\ {\sin \quad \delta \quad y} & 0 & {\cos \quad \delta \quad y} & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix}} \cdot}} \\ {\quad {\begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & {\cos \quad \delta \quad x} & {\sin \quad \delta \quad x} & 0 \\ 0 & {{- \sin}\quad \delta \quad x} & {\cos \quad \delta \quad x} & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix} \cdot \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ \alpha & \beta & \gamma & 1 \end{bmatrix}}} \\ {= \quad \begin{bmatrix} {\cos \quad 2\delta \quad y} & {\sin \quad \delta \quad {x \cdot \sin}\quad 2\delta \quad y} & {{- \cos}\quad \delta \quad {x \cdot \sin}\quad 2\delta \quad y} & 0 \\ {\sin \quad \delta \quad {x \cdot \sin}\quad 2\delta \quad y} & {1 - {2\left( {\sin \quad \delta \quad {x \cdot \cos}\quad \delta \quad y} \right)^{2}}} & {\sin \quad 2\delta \quad {x \cdot \cos^{2}}\delta \quad y} & 0 \\ {{- \cos}\quad \delta \quad {x \cdot \sin}\quad 2\delta \quad y} & {\sin \quad 2\delta \quad {x \cdot \cos^{2}}\delta \quad y} & {1 - {2\left( {\cos \quad \delta \quad {x \cdot \cos}\quad \delta \quad y} \right)^{2}}} & 0 \\ {Kx} & {Ky} & {Kz} & 1 \end{bmatrix}} \end{matrix} & (97) \end{matrix}$

Where

Kx=2αsin² δy−(βsinδx−cosδx)·sin 2δy

Ky=−αsin δx·sin2δy+(2βsin² δx−γsin 2δx)·cos² δy

Kz=αcosδx·sin 2δy−(βsin 2δx−2γcos² δx)·cos² δy

δx=θyz=tan⁻¹(cos θy/cos θz)

δy=θxz=tan⁻¹[cos θx/{square root over ((cos²+L θy+cos²+L θz))}

The symmetric transformation matrix Sg is a matrix for obtaining a symmetric image plane-symmetric to a main image. Therefore, a symmetric image can be obtained from the input source video signal V₂ by making the transformation matrix T (comprising T₀ and P₀) used to obtain a main image from the source video signal V₁ and the symmetric transformation matrix Sg work on the source video signal V₂. That is, by making a four-row four-column symmetric transformation matrix T_(3g) shown by the following expression (98) work on the source video signal V₂. $\begin{matrix} \begin{matrix} {T_{3g} = \quad {T_{0} \cdot {Sg} \cdot P_{0}}} \\ {= \quad {\begin{bmatrix} r_{11} & r_{12} & r_{13} & 0 \\ r_{21} & r_{22} & r_{23} & 0 \\ r_{31} & r_{32} & r_{33} & 0 \\ l_{x} & l_{y} & l_{z} & s \end{bmatrix} \cdot \quad \begin{bmatrix} {\cos \quad 2\delta \quad y} & {\sin \quad \delta \quad {x \cdot \sin}\quad 2\delta \quad y} & {{- \cos}\quad \delta \quad {x \cdot \sin}\quad 2\delta \quad y} & 0 \\ {\sin \quad \delta \quad {x \cdot \sin}\quad 2\delta \quad y} & {1 - {2\left( {\sin \quad \delta \quad {x \cdot \cos}\quad \delta \quad y} \right)^{2}}} & {\sin \quad 2\delta \quad {x \cdot \cos^{2}}\delta \quad y} & 0 \\ {{- \cos}\quad \delta \quad {x \cdot \sin}\quad 2\delta \quad y} & {\sin \quad 2\delta \quad {x \cdot \cos^{2}}\delta \quad y} & {1 - {2\left( {\cos \quad \delta \quad {x \cdot \cos}\quad \delta \quad y} \right)^{2}}} & 0 \\ {Kx} & {Ky} & {Kz} & 1 \end{bmatrix} \cdot \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & P_{z} \\ 0 & 0 & 0 & 1 \end{bmatrix}}} \end{matrix} & (98) \end{matrix}$

Where

Kx=2αsin² δy−(βsinδx−cosδx)·sin 2δy

Ky=−αsin δx·sin2δy+(2βsin² δx−γsin 2δx)·cos² δy

Kz=αcosδx·sin 2δy−(βsin 2δx−2γcos² δx)·cos² δy

δx=θyz=tan⁻¹(cos θy/cos θz)

δy=θxz=tan⁻¹[cos θx/{square root over ((cos²+L θy+cos²+L θz))}

Also in the case of the plane symmetry, the source video signal V₂ and video signal V₆ to be handled are two-dimensional video data. Therefore, a parameter of the symmetric transformation matrix T_(3g) for computing z-axis-directional data from the symmetry transformation matrix T_(3g) shown in the expression (98) is not practically used. Therefore, a three-row three-column transformation matrix T_(3g(33)) excluding the parameter of the third column and third row for computing z-axis-directional data from the symmetry transformation matrix T_(3g) shown in the expression (98) is used for the operation of a read address.

Moreover, in the case of the plane symmetry, image transformation is performed in accordance with the read processing from the frame memory 12A similarly to the case of the line symmetry. Therefore, it is necessary to obtain a point on the frame 12A corresponding to a point on the monitor screen 16. Therefore, in fact, the symmetric transformation matrix ^(T) _(3g(33)) is not necessary but the inverse matrix T_(3g(33)) ⁻¹ of the symmetric transformation matrix T_(3g(33)) is necessary.

In this case, by setting the parameters of the symmetric transformation matrix T_(3g(33)) as shown in the following expression (99), the inverse matrix T_(3g(33)) ⁻¹ of the symmetric transformation matrix T_(3g(33)) is shown by the following expression (100). $\begin{matrix} {T_{3{g{(33)}}} = \begin{bmatrix} a_{11}^{\prime} & a_{12}^{\prime} & a_{13}^{\prime} \\ a_{21}^{\prime} & a_{22}^{\prime} & a_{23}^{\prime} \\ a_{31}^{\prime} & a_{32}^{\prime} & a_{33}^{\prime} \end{bmatrix}} & (99) \\ {T_{3{g{(33)}}}^{- 1} = \begin{bmatrix} b_{11}^{\prime} & b_{12}^{\prime} & b_{13}^{\prime} \\ b_{21}^{\prime} & b_{22}^{\prime} & b_{23}^{\prime} \\ b_{31}^{\prime} & b_{32}^{\prime} & b_{33}^{\prime} \end{bmatrix}} & (100) \\ {{Where}{b_{ij} = \frac{a_{ji}}{\det \left( T_{3{g{(33)}}} \right)}}} & \quad \end{matrix}$

Thus, in the case of the special effect system 10, when plane symmetry is designated by an operator as the symmetric mode of a symmetric image, the CPU 13 obtains the parameters b₁₁′ to b₃₃′ of the inverse matrix T_(3g(33)) ⁻¹ in accordance with a designated reference plane and supplies the parameters b₁₁′ to b₃₃′ to the read address generation circuit 12C. The read address generation circuit 12C calculates the read address (X_(M)′, Y_(M)′) of the frame memory 12A by using the parameters b₁₁′ to b₃₃′ for the symmetric transformation instead of the parameters b₁₁ to b₃₃ of the operation expressions shown in the expressions (15) and (16). Thus, by reading the source video signal V₂ from the frame memory 12A in accordance with the read address (X_(M)′, Y_(M)′), the special effect system 10 can obtain this video signal V₆ of a symmetric image plane-symmetric to a main image.

(3-5-1) Plane symmetry of plane parallel with z-axis

As shown in FIG. 35, A plane-symmetric transformation matrix Sz of a plane Gz passing through a point q (α, β, γ) and parallel with z-axis can be easily obtained by assuming that a three-dimensional coordinate system is projected on a plane z=y and considering the system as line symmetry on the plane z=y. That is, when assuming each angle formed between the plane Gz and the xz plane as θyx, the transformation matrix Sz is shown by the following expression (101) by extending the line-symmetric transformation matrix S1 shown by the expression (67) to a three-dimensional space. $\begin{matrix} \begin{matrix} {{Sz} = \quad {{Lq}^{- 1} \cdot {R\left( {{- \theta}\quad {yx}} \right)} \cdot {Sx} \cdot {R\left( {\theta \quad {yx}} \right)} \cdot {Lq}}} \\ {= \quad {\begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ {- \alpha} & {- \beta} & {- \gamma} & 1 \end{bmatrix} \cdot \begin{bmatrix} {\cos \quad \left( {{- \theta}\quad {yx}} \right)} & {\sin \quad \left( {{- \theta}\quad {yx}} \right)} & 0 & 0 \\ {{- \sin}\quad \left( {{- \theta}\quad {yx}} \right)} & {\cos \quad \left( {{- \theta}\quad {yx}} \right)} & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix} \cdot}} \\ {\quad {\begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & {- 1} & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix} \cdot \begin{bmatrix} {\cos \quad \left( {\theta \quad {yx}} \right)} & {\sin \quad \left( {\theta \quad {yx}} \right)} & 0 & 0 \\ {{- \sin}\quad \left( {\theta \quad {yx}} \right)} & {\cos \quad \left( {\theta \quad {yx}} \right)} & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix} \cdot}} \\ {\quad \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ \alpha & \beta & \gamma & 1 \end{bmatrix}} \\ {= \quad \begin{bmatrix} {\cos \quad \left( {2\theta \quad {yx}} \right)} & {\sin \quad \left( {2\theta \quad {yx}} \right)} & 0 & 0 \\ {\sin \quad \left( {2\theta \quad {yx}} \right)} & {{- \cos}\quad \left( {2\theta \quad {yx}} \right)} & 0 & 0 \\ 0 & 0 & 1 & 0 \\ {K\quad \alpha} & {K\quad \beta} & {2\gamma} & 1 \end{bmatrix}} \end{matrix} & (101) \end{matrix}$

Where

Kα=α·(1−cos(2θyx))−β·sin(2θyx)

Kβ=−α·sin(2θyx)+β·(1+cos(2ηyx))

Therefore, a symmetric transformation matrix T_(3gz) for obtaining a symmetric image from the source video signal V₂ is shown by the following expression (102) by combining the transformation matrix T (comprising T₀ and P₀) for three-dimensional image transformation with the symmetric transformation matrix Sz. $\begin{matrix} \begin{matrix} {T_{3{gz}} = \quad {T_{0} \cdot {Sz} \cdot P_{0}}} \\ {= \quad {\begin{bmatrix} r_{11} & r_{12} & r_{13} & 0 \\ r_{21} & r_{22} & r_{23} & 0 \\ r_{31} & r_{32} & r_{33} & 0 \\ l_{x} & l_{y} & l_{z} & s \end{bmatrix} \cdot}} \\ {\quad {\begin{bmatrix} {\cos \quad \left( {2\theta \quad {yx}} \right)} & {\sin \quad \left( {2\theta \quad {yx}} \right)} & 0 & 0 \\ {\sin \quad \left( {2\theta \quad {yx}} \right)} & {{- \cos}\quad \left( {2\theta \quad {yx}} \right)} & 0 & 0 \\ 0 & 0 & 1 & 0 \\ {K\quad \alpha} & {K\quad \beta} & {2\gamma} & 1 \end{bmatrix} \cdot \quad \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & P_{z} \\ 0 & 0 & 0 & 1 \end{bmatrix}}} \end{matrix} & (102) \end{matrix}$

Where

Kα=α·(1−cos(2θyx))−β·sin(2θyx)

Kβ=−α·sin(2θyx)+β·(1+cos(2θyx))

Thus, it is possible to generate a symmetric image using a plane parallel with z-axis as a reference plane in the CPU 13 by obtaining the parameters b₁₁′ to b₃₃′ of the inverse matrix T_(3gz(33)) ⁻¹ of the transformation matrix T_(3gz(33)) excluding the third column and third row of the symmetric transformation matrix T_(3gz) and supplying the parameters b₁₁′ to b₃₃′ to the read address generation circuit 12C similarly to the case of the above plane symmetry.

(4) Effect file

In this section, an effect file is described in which parameters for image transformation input through the control panel, 14 are entered.

The parameters input through the control panel 14 include a parameter related to a special effect and a parameter for specifying a symmetry yardstick. When these parameters are input, the special effect system 10 forms an effect file according to a predetermined format in a RAM 17 to enter the parameters input to the effect file. The effect file is a data file into which data groups for specifying the processing contents of the image transformation to be applied to the source video signal V₁ or V₂.

FIG. 36 shows the format of an effect file. The structure of the effect file is roughly divided into two types of areas. One is a key frame data area for entering parameters related to a special effect and the other is a symmetric mode area for entering the data for specifying a symmetry yardstick. The key frame data area is divided into areas for entering size data, position data, and rotation data. Parameters for a special effect input through the control panel 14 are respectively entered in their corresponding area.

In other words, the parameters for a special effect input through the control panel 14 include size data, position data, and rotation data. In this case, the size data represents the data for specifying the size of a main image and for specifying the enlargement or reduction processing in the image transformation to be applied to the main image. Moreover, the position data represents the data for specifying a position on the monitor screen 16 where a main image is displayed and for specifying the translation processing in the image transformation to be applied to the main image. Furthermore, the rotation data represents the data for specifying the rotation processing in the image transformation to be applied to a main image.

When parameters for these special effects are entered in an effect file, the CPU 13 computes parameters r₁₁ to r₃₃, l_(x), l_(y), l_(z), P_(z) and s of the transformation matrix T₃₃ of the image transformation to be applied to a main image in accordance with the size data, position data, and rotation data entered in the effect file, computes the matrix parameters b₁₁ to b₃₃ of the inverse matrix T₃₃ ⁻¹ of the transformation matrix T₃₃ in accordance with the parameters, and supplies the parameters b₁₁ to b₃₃ to the read address generation, circuit 11C.

Moreover, in the case of the special effect system 10, several symmetric modes to be relatively easily used are previously prepared so that a desired symmetric mode can be selected and specified from the prepared symmetric modes. Therefore, areas for entering the prepared symmetric modes are prepared for the symmetric mode area of an effect file. Therefore, when a symmetric mode is designated by an operator, the start information for starting the designated symmetric mode is entered in a corresponding area. Specifically, when origin symmetry is designated, the start information such as “on” for starting the origin symmetry is entered in an area for origin symmetry and the non-start information such as “off” is entered in not-designated symmetric mode areas. The prepared symmetric modes include origin-symmetric mode, x-axis-symmetric mode, y-axis-symmetric mode, z-axis-symmetric mode, xy-plane-symmetric mode, yz-plane-symmetric mode, and xz-symmetric mode as shown in FIG. 36. In this connection, symmetric transformation matrixes for these symmetric modes are previously calculated and stored in the memory 17.

When the start information such as “on” for starting a symmetric mode is entered in a symmetric mode area, the CPU 13 reads the symmetric transformation matrix Sp, Sq, S1, or Sg of the symmetric mode in which “on” is entered from the memory 17, obtains the matrix parameters b₁₁′ to b₃₃′ of transformation matrix T_(2p(33)) ⁻¹, T_(3p(33)) ⁻¹, T₂₁₍₃₃₎ ⁻¹, T₃₁₍₃₃₎ ⁻¹, or T_(3g(33)) ⁻¹ for obtaining a symmetric image in accordance with the read symmetric transformation matrix Sp, Sq, S1, or Sg and the matrix parameter of the transformation matrix T₃₃ obtained from a parameter for a special effect, and supplies the matrix parameters b₁₁′ to b₃₃′ to the read address generation circuit 11C.

The special effect system 10 has a two-channel structure capable of simultaneously processing a main image and a symmetric image as shown in FIG. 3. Therefore, when receiving special-effect parameters (size data, position data, and rotation data) to be applied to a main image, the parameters are entered in an effect file and simultaneously, an effect file for a symmetric image is also automatically formed and the size data, position data, and rotation data same as those of the main image are entered in the automatically formed effect file. Thereby, to generate a symmetric image, only by entering a symmetric mode in the automatically-generated effect file for a symmetric image, it is possible to easily generate a symmetric image without entering size data, position data, and rotation data by an operator. In this connection, by setting every symmetric mode to “off” in the effect file for a symmetric image, no symmetric image is generated. Even in this case, however, the effect file for a symmetric image is not erased so that a symmetric image can be generated any time.

In the case of the special effect system 10, as shown in FIG. 37, a plurality of data groups (hereafter referred to as key frames) specifying the above image processing contents can be entered in ones effect file and thereby, an image whose shapes and/or positions are successively changed can be generated in accordance with the entered key frames. For example, by entering three key frames KF1, KF2, and KF3 in which only position data is different in a main-image effect file as shown in FIG. 37, it is possible to obtain an image 6 in which a main image 5 successively continuously moves from the position specified by the key frame KF1 to the position specified by the key frame KF2 and then, successively continuously moves from the position specified by the key frame KF2 to the position specified by the key frame KF3.

Thus, also by entering a plurality of key frames in the main-image effect file, a symmetric-image effect file having the same key frames is automatically generated as described above. That is, by entering three key frames KF1, KF2, and KF3 in a main-image effect file as shown in FIG. 37, a symmetric-image effect file having key frames KF1′, KF2′, and KF3′ having the same data contents as those of the key frames KF1, KF2, and KF3 is automatically generated as shown in FIG. 39. Therefore, it is possible to easily obtain an image in which a symmetric image successively moves on symmetric positions in accordance with the movement of a main image only by entering the symmetry yardstick of the image in an effect file. For example, by entering a y-axis-symmetric mode in a symmetric-image effect file so as to be “on” as shown in FIG. 39, it is possible to easily obtain an image 6 in which a symmetric image 5A successively continuously moves on y-axis-symmetric positions of a main image 5 in accordance with the movement of the main image 5.

In this connection, the CPU 13 calculates size data, position data, and rotation data to be continuously changed in accordance with the data of the key frames KF1, KF2, and KF3 for the spaces between the key frames KF1 and KF2 and the space between the key frames KF2 and KF3 and calculates a transformation matrix for image transformation in accordance with the calculated size data, position data, and rotation data. Thereby, an operator only needs to enter a key frame only for a key point portion for deforming and/or moving an image and therefor, he can easily obtain a continuously-changing image without entering every key frame related to deformation and/or movement of the image.

In this section, a case is described in which an origin-symmetric mode, x-axis-symmetric mode, y-axis-symmetric mode, z-axis-symmetric mode, xy-plane-symmetric mode, yz-plane-symmetric mode, and xz-plane-symmetric mode are prepared as symmetric modes and a desired symmetry yardstick is selected out of the symmetric modes and entered. In the case of the special effect system 10, however, it is also possible to obtain a symmetric image by specifying an optional point, line, or plane as a symmetry yardstick.

To specify an optional point, line, or plane as a symmetry yardstick, by inputting a parameter for specifying the point, line, or plane serving as the symmetry yardstick, the parameter is entered in an effect file. Therefore, the CPU 13 reads the parameter from the effect file and computes the inverse matrix parameters b₁₁′ to b₃₃′ of a symmetric transformation matrix. Thereby, only by inputting a parameter for specifying an optional point, line, or plane, it is possible to easily obtain a symmetric image using the optional point, line, or plane as a symmetry yardstick.

(5) Description of operations and advantages of special effect system

In the case of the special effect system 10 having the above structure, to apply three-dimensional image transformation, that is, a special effect to the input source video signal V₁, an operator inputs desired special-effect parameters (size data, position data, and rotation data) through the control panel 14. The CPU 13 of the special effect system 10 enters the special-effect parameters in an effect file to compute the matrix parameters b₁₁ to b₃₃ of the transformation matrix T₃₃ ⁻¹ for image transformation in accordance with the special-effect parameters and supplies the parameters b₁₁ to b₃₃ to the read address generation circuit 11C of the image processing section 11.

The read address generation circuit 11C generates the read address (X_(M), Y_(M)) for applying a designated special effect to the source video signal V₁ in accordance with the matrix parameters b₁₁ to b₃₃ and the screen address (X_(S), Y_(S)) supplied from the screen address generation circuit 18 and supplies the read address (X_(M), Y_(M)) to the frame memory 11A of the first image processing section 11.

The frame memory 11A successively stores the input source video signal V₁ in internal storage areas and applies three-dimensional image transformation to the source video signal V₁ by reading video data from a storage area designated by the read address (X_(M), Y_(M)) and outputting the data, and thereby generates the video signal V₃ of a main image to which a special effect designated by an operator is applied. The video signal V₃ of the main image is interpolated through the interpolation circuit 11D and thereafter, output to the mixer 20 as the video signal V₄.

To generate a symmetric image point-, line-, or plane-symmetric to a main image from the source Video signal V₂, an operator inputs the data for specifying a symmetry yardstick through the control panel 14. The CPU 13 calculates the symmetric transformation matrixes Sp, Sq, S1, or Sg in accordance with the data for specifying the symmetry yardstick and the matrix parameter of the transformation matrix T₃₃ of three-dimensional image transformation, computes the matrix parameters b₁₁′ to b₃₃′ of the inverse matrix T_(2p(33)) ⁻¹, T_(3p(33)) ⁻¹, T₂₁₍₃₃₎ ⁻¹, T₃₁₍₃₃₎ ⁻¹, or T_(3g(33)) ⁻¹ of the symmetric transformation matrix, and supplies the matrix parameters b₁₁′ to b₃₃′ to the read address generation circuit 12C of the second image processing section 12.

The read address generation circuit 12C generates the read address (X_(M)′, Y_(M)′) for generating a designated symmetric image in accordance with the matrix parameters b₁₁′ to b₃₃′ the screen address (Xs, Yx) supplied from the screen address generation circuit 18 and supplies the read address (X_(M)′, Y_(M)′) to the frame memory 12A of the second image processing section 12.

The frame memory 12A successively stores the input source video signal V₂ in its internal storage areas, applies symmetric image transformation to the source video signal V₂ by reading video data from the storage area specified by the read address (X_(M)′, Y_(M)′) and outputting it, and thereby generates the video signal V₅ of a symmetric image for a main image based on the point, line, or plane designated by an operator. The symmetric-image video signal V₅ is interpolated through the interpolation circuit 12D and thereafter, output to the mixer 20 as the video signal V₆.

The mixer 20 mixes the main-image video signal V₄ with the symmetric-image video signal V₆ to generate a mixed video signal V_(MIX) and the mixer 21 mixes the mixed video signal V_(MIX) with a background video signal V_(BK). Thereby, a video signal V_(OUT) comprising a main image and a symmetric image point-, line-, or plane-symmetric to the main image is generated.

Thus, in the case of the special effect system 10, a symmetric image point-, line-, or plane-symmetric to a main image to which three-dimensional image transformation is applied can be generated only by designating a point, line, or plane serving as the symmetry yardstick of the symmetric image without calculating and inputting a symmetric position by an operator as ever. Moreover, it is possible to generate a more accurate symmetric image because an accurate symmetric position can be calculated in accordance with a designated point, line, or plane compared to the conventional case of designating a symmetric position by eye measure while viewing a screen by using an input unit such as a track ball, joy stick, or mouse.

Furthermore, in the case of the special effect system 10, even when entering a plurality of key frames to continuously move an image, the effect file of a symmetric image having key frames for a main image is automatically generated by entering the key frames respectively in an effect file. Therefore, only by entering a symmetry yardstick in the effect file of the symmetric image, it is possible to easily obtain a symmetric image successively continuously moving on symmetric positions in accordance with the movement of the main image and moreover easily obtain the symmetric image without designating a symmetric position each time as ever.

According to the above structure, the matrix parameters b₁₁′ to b₃₃′ for symmetric image transformation are computed in accordance with a point, line, or plane serving as a designated symmetry yardstick and made to work on the read address (X_(M)′, Y_(M)′) from the frame memory 12A and generate a symmetric image. Therefore, it is possible to easily generate an accurate symmetric image only by designating a point, line, or plane serving as a symmetry yardstick and generate an accurate symmetric image without requesting an operator to perform troublesome operations.

In the above description, a case is described in which an origin-symmetric mode, x-axis-symmetric mode, y-axis-symmetric mode, z-axis-symmetric mode, xy-plane-symmetric mode, yz-plane-symmetric mode, and xz-plane-symmetric mode are prepared as symmetric modes to be relatively easily used. However, it is also possible to use symmetric modes other than the above modes as the prepared symmetric modes. In short, by preparing symmetric modes to be relatively easily used and storing the symmetric transformation matrixes of the symmetric modes in a memory, an operator can easily obtain a symmetric image only by selecting the prepared symmetric modes.

Moreover, when preparing the symmetric modes to be relatively easily used, it is possible to easily designate a desired symmetric mode by providing exclusive keys for specifying the symmetric modes such as an origin-symmetric mode key, x-axis-symmetric mode key, y-axis-symmetric mode key, z-axis-symmetric mode key, xy-plane-symmetric mode key, yz-plane-symmetric mode key, and xz-plane-symmetric mode key for the control panel 14.

Furthermore, in the case of the above description, a case is described in which the first source video signal V₁ is input to the first image processing section 11 and the second source video signal V₂ different from the first source video signal V₁ is input to the second image processing section 12. However, it is also possible to input the same source video signal to the first and second image processing sections 11 and 12. Thus, it is possible to obtain a complete symmetric image including image contents. For example, by inputting the same source video signal and designating the y-axis-symmetric mode, it is possible to obtain a symmetric image seemingly reflected by a mirror.

INDUSTRIAL APPLICABILITY

The present invention can be used to generate a television broadcasting signal to which a special effect is applied in a broadcasting station. 

What is claimed is:
 1. A special effect system for applying image transformation to an input source video signal, comprising: first image processing means for generating the video signal of a main image by performing an image transformation on an input first source video signal in accordance with a first image transformation matrix; and second image processing means for generating the video signal of a symmetric image point-symmetric, line-symmetric, or plane-symmetric to said main image by receiving a second source video signal which is the same as or different from said first source video signal and performing an image transformation on said second source video in accordance with a transformation matrix obtained by combining the image transformation matrix used to generate said main image with a symmetric image transformation matrix for a point, line, or plane specified as a symmetry yardstick.
 2. A special effect system for applying image transformation to an input source video signal, comprising: control means for computing a first matrix parameter for image transformation in accordance with a specified special-effect parameter and computing a second matrix parameter for obtaining a point-symmetric, line-symmetric, or plane-symmetric image in accordance with said special effect parameter and the specified point, line, or plane serving as a symmetry yardstick; first read address generation means for receiving said first matrix parameter and generating a first read address for image transformation in accordance with said first matrix parameter; second read address generation means for receiving said second matrix parameter and generating a second read address for obtaining a point-symmetric, line-symmetric, or plane-symmetric image in accordance with said second matrix parameter; first memory means for successively storing an input first source video signal in its internal storage areas and successively reading said first source video signal from a storage area specified by said first read address and thereby, applying image transformation specified by said special effect parameter to said first source video signal to generate the video signal of a main image; and second memory means for receiving a second source video signal same as or different from said first source video signal, successively storing said second source video signal in its internal storage areas and successively reading said second source video signal from a storage area specified by said second read address, and thereby generating the video signal of a symmetric image point-symmetric, line-symmetric, or plane-symmetric to said main image.
 3. The special effect system according to claim 2, wherein said control means, when the point, line, or plane prepared as said symmetry yardstick is specified, reads a symmetric transformation matrix corresponding to said specified symmetry yardstick from third memory means and computes said second matrix parameter in accordance with said symmetric transformation matrix.
 4. The special effect system according to claim 2, wherein said control means computes said first matrix parameter in accordance with said special effect parameter entered in the effect file of a main image and computes said second matrix parameter in accordance with said special effect parameter entered in the effect file of a symmetric image and said symmetry yardstick.
 5. The special effect system according to claim 4, wherein said control means, when entering said special effect parameter in the effect file of said main image, forms a symmetric-image effect file having said special effect parameter in accordance with the effect file of said main image and, when said symmetry yardstick is specified, enters said specified symmetry yardstick in the effect file of said symmetric image.
 6. The special effect system according to claim 4, wherein said control means, when a plurality of said special effect parameters are entered in the effect file of said main image, computes said first matrix parameter in which positions and/or shapes of said main image are continuously changed in accordance with said entered special-effect parameters.
 7. The special effect system according to claim 4, wherein said control means, when a plurality of said special effect parameters are entered in the effect file of said symmetric image,. computes said second matrix parameter in which positions and/or shapes of said symmetric image are continuously changed in accordance with said entered special effect parameters and said symmetry yardstick.
 8. An image processing method comprising the steps of: performing an image transformation on a first input source video signal in accordance with a first image transformation matrix, thereby generating the video signal of a main image; and performing an image transformation on a second source video signal which is the same as or different than said first source video signal, in accordance with a transformation matrix obtained by combining the image transformation matrix used to generate said main image with a symmetric image transformation matrix for a point, line, or plane specified as a symmetry yardstick, thereby generating the video signal of a symmetric image point-symmetric, line-symmetric, or plane-symmetric to said main image.
 9. A symmetric image generation method for generating a symmetric image point-symmetric, line-symmetric, or plane-symmetric to a main image generated by applying image transformation to a video signal, comprising the step of: applying image transformation to an input source video signal in accordance with a transformation matrix obtained by combining the image transformation matrix used to generate said main image with a symmetric transformation matrix for the point, line, or plane specified as a symmetry yardstick and thereby, generating a symmetric image point-symmetric, line-symmetric, or plane-symmetric to said main image.
 10. Image processing apparatus for processing first and second video signals, the apparatus comprising: first processing means for performing a first video transformation on said first video signal in accordance with a first transformation matrix to generate a first transformed video signal, wherein said first transformation matrix comprises a three-dimensional transformation matrix having specified three-dimensional effect parameters; second processing means for performing a second video transformation on said second video signal in accordance with a second transformation matrix to generate a second transformed video signal, wherein said second transformation matrix comprises said three-dimensional transformation matrix and a symmetric transformation matrix having symmetric parameters; and generating means for generating said three-dimensional effect parameters and said symmetric parameters so that said second transformed video signal represents a symmetric image to an image of said first transformed video signal.
 11. Image processing method for processing first and second video signals, comprising the steps of: performing a first video transformation on said first video signal in accordance with a first transformation matrix to generate a first transformed video signal, wherein said first transformation matrix comprises a three-dimensional transformation matrix having specified three-dimensional effect parameters; performing a second video transformation on said second video signal in accordance with a second transformation matrix to generate a second transformed video signal, wherein said second transformation matrix comprises said three-dimensional transformation matrix and a symmetric transformation matrix having symmetric parameters; and generating said three-dimensional effect parameters and said symmetric parameters so that said second transformed video signal represents a symmetric image to an image of said first transformed video signal.
 12. Video effect apparatus for generating a video effect on first video and second video signals to generate a processed video signal, the apparatus comprising: means for performing a first video transformation on said first video signal in accordance with a three-dimensional transformation matrix to generate a first transformed video signal; means for performing said first video transformation on a first key signal corresponding to said first video signal in accordance with said three-dimensional transformation matrix to generate a first transformed key signal; means for performing a second video transformation on said second video signal in accordance with said three-dimensional transformation matrix and a symmetric transformation matrix to generate a second transformed video signal; means for performing a second video transformation on a second key signal corresponding to said second video signal in accordance with said three-dimensional transformation matrix and said symmetric transformation matrix to generate a second transformed key signal; and means for combining said first and second transformed video signals by using said first and second transformed key signals to generate said processed video signal.
 13. A method for generating a video effect on first and second video signals to generate a processed video signal, comprising the steps of: performing a first video transformation on said first video signal in accordance with a three-dimensional transformation matrix to generate a first transformed video signal; performing said first video transformation on a first key signal corresponding to said first video signal in accordance with said three-dimensional transformation matrix to generate a first transformed key signal; performing a second video transformation on said second video signal in accordance with said three-dimensional transformation matrix and a symmetric transformation matrix to generate a second transformed video signal; performing a second video transformation on a second key signal corresponding to said second video signal in accordance with said three-dimensional transformation matrix and said symmetric transformation matrix to generate a second transformed key signal; and combining said first and second transformed video signals by using said first and second transformed key signals to generate said processed video signal. 